Levy matters I: Foundations, trees and numerical issues in finance

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 2001 Lévy in Stan...

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

2001

Lévy in Stanford (Permission granted by G.L. Alexanderson) “Lévy Matters” is a subseries of the Springer Lecture Notes in Mathematics, devoted to the dissemination of important developments in the area of Stochastics that are rooted in the theory of Lévy processes. Each volume will contain state-of-the-art theoretical results as well as applications of this rapidly evolving field, with special emphasis on the case of discontinuous paths. Contributions to this series by leading experts will present or survey new and exciting areas of recent theoretical developments, or will focus on some of the more promising applications in related fields. In this way each volume will constitute a reference text that will serve PhD students, postdoctoral researchers and seasoned researchers alike. Editors Ole E. Barndorff-Nielsen Thiele Centre for Applied Mathematics in Natural Science Department of Mathematical Sciences ˚ Arhus University, Ny Munkegade 118 ˚ DK-8000 Arhus C, Denmark [email protected]

Jean Jacod Institut de Mathématiques de Jussieu CNRS-UMR 7586 Université Paris 6 – Pierre et Marie Curie Case courrier 188, 4 Place Jussieu 75252 Paris Cedex 05, France [email protected]

Jean Bertoin Laboratoire de Probabilités et Modèles Aléatoires Université Paris 6 – Pierre et Marie Curie Case courrier 188, 4 Place Jussieu 75252 Paris Cedex 05, France [email protected]

Claudia Klüppelberg TUM Institute for Advanced Study & Zentrum Mathematik Technische Universität München Boltzmannstraße 3 85747 Garching bei München, Germany [email protected]

Managing Editors Vicky Fasen Zentrum Mathematik Technische Universität München Boltzmannstraße 3 85747 Garching bei München, Germany [email protected]

Robert Stelzer TUM Institute for Advanced Study & Zentrum Mathematik Technische Universität München Boltzmannstraße 3 85747 Garching bei München, Germany [email protected]

The volumes in this subseries are published under the auspices of the Bernoulli Society.

Thomas Duquesne • Oleg Reichmann Ken-iti Sato • Christoph Schwab

Lévy Matters I Recent Progress in Theory and Applications: Foundations, Trees and Numerical Issues in Finance With a Short Biography of Paul Lévy by Jean Jacod Editors: Ole E. Barndorff-Nielsen Jean Bertoin Jean Jacod Claudia Klüppelberg

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Thomas Duquesne Université Paris 6 – Pierre et Marie Curie Laboratoire de Probabilités et Modèles Aléatoires Case courrier 188, 4 Place Jussieu 75252 Paris CX 05 France [email protected]

Oleg Reichmann ETH Zürich Seminar für Angewandte Mathematik Rämistrasse 101 8092 Zürich Switzerland [email protected]

Ken-iti Sato Hachiman-yama 1101-5-103 Tenpaku-Ku Nagoya 468-0074 Japan [email protected]

Christoph Schwab ETH Zürich Seminar für Angewandte Mathematik Rämistrasse 101 8092 Zürich Switzerland [email protected]

ISBN: 978-3-642-14006-8 e-ISBN: 978-3-642-14007-5 DOI: 10.1007/978-3-642-14007-5 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010933508 Mathematics Subject Classification (2010): 60G51, 60E07, 60J80, 45K05, 65N30, 28A78, 60H05, 60G57, 60J75 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. c Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach Cover photograph: Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper springer.com

Preface

Over the past 10-15 years, we have seen a revival of general Lévy processes theory as well as a burst of new applications. In the past, Brownian motion or the Poisson process have been considered as appropriate models for most applications. Nowadays, the need for more realistic modelling of irregular behaviour of phenomena in nature and society like jumps, bursts, and extremes has led to a renaissance of the theory of general Lévy processes. Theoretical and applied researchers in fields as diverse as quantum theory, statistical physics, meteorology, seismology, statistics, insurance, finance, and telecommunication have realised the enormous flexibility of Lévy models in modelling jumps, tails, dependence and sample path behaviour. Lévy processes or Lévy driven processes feature slow or rapid structural breaks, extremal behaviour, clustering, and clumping of points. Tools and techniques from related but disctinct mathematical fields, such as point processes, stochastic integration, probability theory in abstract spaces, and differential geometry, have contributed to a better understanding of Lévy jump processes. As in many other fields, the enormous power of modern computers has also changed the view of Lévy processes. Simulation methods for paths of Lévy processes and realisations of their functionals have been developed. Monte Carlo simulation makes it possible to determine the distribution of functionals of sample paths of Lévy processes to a high level of accuracy. This development of Lévy processes was accompanied and triggered by a series of Conferences on Lévy Processes: Theory and Applications. The First and Second Conferences were held in Aarhus (1999, 2002), the Third in Paris (2003), the Fourth in Manchester (2005), and the Fifth in Copenhagen (2007). To show the broad spectrum of these conferences, the following topics are taken from the announcement of the Copenhagen conference: • • • • • • • •

Structural results for Lévy processes: distribution and path properties Lévy trees, superprocesses and branching theory Fractal processes and fractal phenomena Stable and infinitely divisible processes and distributions Applications in finance, physics, biosciences and telecommunications Lévy processes on abstract structures Statistical, numerical and simulation aspects of Lévy processes Lévy and stable random fields. v

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Preface

At the Conference on Lévy Processes: Theory and Applications in Copenhagen the idea was born to start a series of Lecture Notes on Lévy processes to bear witness of the exciting recent advances in the area of Lévy processes and their applications. Its goal is the dissemination of important developments in theory and applications. Each volume will describe state of the art results of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts will present new exciting fields, or surveys of recent developments, or focus on some of the most promising applications. Despite its special character, each article is written in an expository style, normally with an extensive bibliography at the end. In this way each article makes an invaluable comprehensive reference text. The intended audience are PhD and postdoctoral students, or researchers, who want to learn about recent advances in the theory of Lévy processes and to get an overview of new applications in different fields. Now, with the field in full flourish and with future interest definitely increasing it seemed reasonable to start a series of Lecture Notes in this area. The present volume is the first in the series, and future volumes will appear over time under the common name “Lévy Matters”, in tune with the developments in the field. “Lévy Matters” will appear as a subseries of the Springer Lecture Notes in Mathematics, thus ensuring wide dissemination of the scientific material. The expository articles in this first volume have been chosen to reflect the broadness of the area of Lévy processes. The first article by Ken-iti Sato characterises extensions of the class of selfdecomposable distributions on Rd . They are given as two families each with two continuous parameters of classes of distributions of improper stochastic integrals limt→∞ 0t f (s)dXs for appropriate non-random functions f and Lévy processes X. Many known classes appear as limiting cases in some parameters: the Thorin class, the Goldie-Steutel-Bondesson class, and the class of completely selfdecomposable distributions. Moreover, the theory of fractional integrals of measures is built. The second article by Thomas Duquesne discusses Hausdorff and packing measures of stable trees. Stable trees are a special class of Lévy trees, which form a class of random compact metric spaces, and were introduced by Le Gall and Le Jan (1998) as the genealogy of continuous state branching processes. It is shown that level sets of stable trees are the sets of points situated at a given distance from the root. In contrast to Brownian trees, for non-Brownian stable trees there is no exact packing measure for level sets, i.e. the sets of points situated at a given distance from the root. The third (and last) article by Oleg Reichmann and Christoph Schwab presents numerical solutions to Kolmogorov equations, which arise for instance in financial engineering, when Lévy or additive processes model the dynamics of the risky assets. Solution algorithms based on wavelet representations for the Dirichlet and free boundary problems connected to barrier and American style contracts are presented. Lévy copulas are used for a systematic construction of parametric multivariate Feller-Lévy processes. Numerical aspects of the implementation and Monte Carlo path simulation techniques are addressed.

Preface

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We take the possibility to acknowledge the very positive collaboration with the relevant Springer staff and the Editors of the LN Series, and the (anonymous) referees of the three articles. We hope that the readers of this and subsequent volumes enjoy learning about the high potential of Lévy processes in theory and applications. Researchers with ideas for contributions to further volumes in the Lévy Matters series are invited to contact any of the Editors with proposals or suggestions. June 2010

Ole E. Barndorff-Nielsen (Aarhus) Jean Bertoin (Paris) Jean Jacod (Paris) Claudia Klüppelberg (Munich)

Contents

Fractional Integrals and Extensions of Selfdecomposability .. . . . . . . . . . . . . . . . . Ken-iti Sato 1 Introduction .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.1 Characterizations of Selfdecomposable Distributions. .. . . . . . . . . . . . . . . . . 1.2 Nested Classes of Multiply Selfdecomposable Distributions .. . . . . . . . . . 1.3 Continuous-Parameter Extension of Multiple Selfdecomposability .. . . 1.4 Stable Distributions and the Class L∞ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.5 Fractional Integrals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.6 The Classes Kp,α and Lp,α Generated by Stochastic Integral Mappings .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.7 Remarkable Subclasses of ID . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2 Fractional Integrals and Monotonicity of Order p > 0 . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2 One-to-One Property .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3 More Properties and Examples .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3 Preliminaries in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 Lévy–Khintchine Representation of Infinitely Divisible Distributions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 Radial and Spherical Decompositions of σ -Finite Measures on Rd . . . . 3.3 Weak Mean of Infinitely Divisible Distributions . . . . . . .. . . . . . . . . . . . . . . . . 3.4 Stochastic Integral Mappings of Infinitely Divisible Distributions . . . . . 3.5 Transformation of Lévy Measures . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4 First Two-Parameter Extension Kp,α of the Class L of Selfdecomposable Distributions .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1 Φf and ΦfL for f = ϕα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.2 Φ¯ p,α and Φ¯ p,L α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.3 Range of Φ¯ p,L α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.4 Classes Kp,α , Kp,0 α , and Kp,e α . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5 One-Parameter Subfamilies of {Kp,α } . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.1 Kp,α , K0p,α , and Kep,α for p∈(0, ∞) with Fixed α . . . . . . . .. . . . . . . . . . . . . . . . . 5.2 Kp,α , K0p,α , and Kep,α for α ∈(−∞, 2) with Fixed p . . . . . .. . . . . . . . . . . . . . . . .

1 2 2 4 4 5 6 8 10 11 11 15 19 26 26 27 29 31 36 37 37 41 45 47 57 57 65

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Contents

6 Second Two-Parameter Extension Lp,α of the Class L of Selfdecomposable Distributions .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.1 Λp,α and Λp,L α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.2 Range of Λp,L α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.3 Classes Lp,α , L0p,α , and Lep,α . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.4 Relation Between Kp,α and Lp,α . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7 One-Parameter Subfamilies of {Lp,α }. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.1 Lp,α , L0p,α , and Lep,α for p∈(0, ∞) with fixed α . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.2 Lp,α , L0p,α , and Lep,α for α ∈(−∞, 2) with Fixed p . . . . . .. . . . . . . . . . . . . . . . . References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

68 68 73 74 77 78 78 87 89

Packing and Hausdorff Measures of Stable Trees . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 93 Thomas Duquesne 1 Introduction .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 93 2 Notation, Definitions and Preliminary Results . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .102 2.1 Hausdorff and Packing Measures on Metric Spaces . . .. . . . . . . . . . . . . . . . .102 2.2 Height Processes and Lévy Trees . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .103 2.3 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .113 3 Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .117 3.1 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .117 3.2 Proof of Proposition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .121 3.3 Proof of Proposition 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .122 3.4 Proof of Theorems 1.6 and 1.10 .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .123 References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .135 Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .137 Oleg Reichmann and Christoph Schwab 1 Introduction .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .138 2 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .140 2.1 Time-Homogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .140 2.2 Time-Inhomogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .144 3 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .145 4 Multivariate Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .148 4.1 Copula Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .148 4.2 Sector Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .151 4.3 A Class of Admissible Market Models . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .154 5 Variational PIDE Formulations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .156 5.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157 5.2 American Options .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .160 5.3 Greeks . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .164 6 Wavelets . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168 6.1 Spline Wavelets on an Interval . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .169 6.2 Tensor Product Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 6.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .174 6.4 Wavelet Compression .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .177

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7 Computational Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .178 7.1 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .178 7.2 Numerical Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .179 8 Alternative Pricing Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .181 8.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .181 8.2 Fourier Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .183 9 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 9.1 Univariate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .186 9.2 Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .190 List of Symbols .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .192 References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .193 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .197

A Short Biography of Paul Lévy

The first volume of the series “Lévy Matters” would not be complete without a short sketch about the life and mathematical achievements of the mathematician whose name has been borrowed and used here. This is more a form of tribute to Paul Lévy, who not only invented what we call now Lévy processes, but also is in a sense the founder of the way we are now looking at stochastic processes, with emphasis on the path properties. Paul Lévy was born in 1886, and lived until 1971. He studied at the Ecole Polytechnique in Paris, and was soon appointed as professor of mathematics in the same institution, a position that he held from 1920 to 1959. He started his career as an analyst, with 20 published papers between 1905 (he was then 19 years old) and 1914, and he became interested in probability by chance, so to speak, when asked to give a series of lectures on this topic in 1919 in that same school: this was the starting point of an astounding series of contributions in this field, in parallel with a continuing activity in functional analysis. Very briefly, one can mention that he is the mathematician who introduced characteristic functions in full generality, proving in particular the characterisation theorem and the first “Lévy’s theorem” about convergence. This naturally led him to study more deeply the convergence in law with its metric, and also to consider sums of independent variables, a hot topic at the time: Paul Lévy proved a form of the 0-1 law, as well as many other results, for series of independent variables. He also introduced stable and quasi-stable distributions, and unravelled their weak and/or strong domains of attractions, simultaneously with Feller. Then we arrive at the book “Théorie de l’addition des variables aléatoires”, published in 1937, and in which he summarizes his findings about what he called “additive processes” (the homogeneous additive processes are now called Lévy processes, but he did not restrict his attention to the homogeneous case). This book contains a host of new ideas and new concepts: the decomposition into the sum of jumps at fixed times and the rest of the process; the Poissonian structure of the jumps for an additive process without fixed times of discontinuities; the “compensation” of those jumps so that one is able to sum up all of them; the fact that the remaining continuous part is Gaussian. As a consequence, he implicitly gave the formula providing the form of all additive processes without fixed discontinuities, now called the Lévy-Itô Formula, and he proved the Lévy-Khintchine formula for the characteristic xiii

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A Short Biography of Paul Lévy

functions of all infinitely divisible distributions. But, as fundamental as all those results are, this book contains more: new methods, like martingales which, although not given a name, are used in a fundamental way; and also a new way of looking at processes, which is the “pathwise” way: he was certainly the first to understand the importance of looking at and describing the paths of a stochastic process, instead of considering that everything is encapsulated into the distribution of the processes. This is of course not the end of the story. Paul Lévy undertook a very deep analysis of Brownian motion, culminating in his book “Processus stochastiques et mouvement brownien” in 1948, completed by a second edition in 1965. This is a remarkable achievement, in the spirit of path properties, and again it contains so many deep results: the Lévy modulus of continuity, the Hausdorff dimension of the path, the multiple points, the Lévy characterisation theorem. He introduced local time, proved the arc-sine law. He was also the first to consider genuine stochastic integrals, with the area formula. In this topic again, his ideas have been the origin of a huge amount of subsequent work, which is still going on. It also laid some of the basis for the fine study of Markov processes, like the local time again, or the new concept of instantaneous state. He also initiated the topic of multi-parameter stochastic processes, introducing in particular the multi-parameter Brownian motion. As should be quite clear, the account given here does not describe the whole of Paul Lévy’s mathematical achievements, and one can consult for many more details the first paper (by Michel Loève) published in the first issue of the Annals of Probability (1973). It also does not account for the humanity and gentleness of the person Paul Lévy. But I would like to end this short exposition of Paul Lévy’s work by hoping that this new series will contribute to fulfilling the program, which he initiated. Jean Jacod (Paris)

Fractional Integrals and Extensions of Selfdecomposability Ken-iti Sato

Abstract After characterizations of the class L of selfdecomposable distributions on Rd are recalled, the classes K p,α and L p,α with two continuous parameters 0 < p < ∞ and −∞ < α < 2 satisfying K1,0 = L1,0 = L are introduced as extensions of the class L. They are defined as the classes of distributions of improper (ρ ) stochastic integrals 0∞− f (s)dXs , where f (s) is an appropriate non-random func(ρ ) tion and Xs is a Lévy process on Rd with distribution ρ at time 1. The description of the classes is given by characterization of their Lévy measures, using the notion of monotonicity of order p based on fractional integrals of measures, and in some cases by addition of the condition of zero mean or some weaker conditions that are newly introduced – having weak mean 0 or having weak mean 0 absolutely. The class Ln,0 for a positive integer n is the class of n times selfdecomposable distributions. Relations among the classes are studied. The limiting classes as p → ∞ are analyzed. The Thorin class T , the Goldie–Steutel–Bondesson class B, and the class L∞ of completely selfdecomposable distributions, which is the closure (with respect to convolution and weak convergence) of the class S of all stable distributions, appear in this context. Some subclasses of the class L∞ also appear. The theory of fractional integrals of measures is built. Many open questions are mentioned. AMS Subject Classification 2000: Primary: 60E07, 60H05 Secondary: 26A33, 60G51, 62E10, 62H05 Keywords Class L · Class L∞ · Completely monotone · Fractional integral · Improper stochastic integral · Infinitely divisible · Lévy measure · Lévy process · Lévy-Khintchine triplet · Monotone of order p · Multiply selfdecomposable · Radial decomposition · selfdecomposable · Spherical decomposition · Weak mean K. Sato () Hachiman-yama 1101-5-103, Tenpaku-Ku, Nagoya, 468-0074 Japan e-mail: [email protected]

T. Duquesne et al., Lévy Matters I: Recent Progress in Theory and Applications: Foundations, Trees and Numerical Issues in Finance, Lecture Notes in Mathematics 2001, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-14007-5 1,

1

2

K. Sato

1 Introduction 1.1 Characterizations of Selfdecomposable Distributions A distribution μ on Rd is called infinitely divisible if, for each positive integer n, there is a distribution μn such that

μ = μn ∗ μn ∗ · · · ∗ μn , n

where ∗ denotes convolution. The class of infinitely divisible distributions on Rd is denoted by ID = ID(Rd ). Let Cμ (z), z ∈ Rd , denote the cumulant function of μ ∈ ID, that is, the unique complex-valued continuous function on Rd (z) of μ is expressed as with Cμ (0) = 0 such that the characteristic function μ C (z) μ (z) = e μ . If μ ∈ ID, then Cμ (z) is expressed as 1 Cμ (z) = − z, Aμ z + 2

Rd

(eiz,x − 1 − iz, x1{|x|≤1}(x))νμ (dx) + iγμ , z. (1.1)

Here z, x is the canonical inner product of z and x in Rd , |x| = x, x1/2 , 1{|x|≤1} is the indicator function of the set {|x| ≤ 1}, Aμ is a d × d symmetric nonnegativedefinite matrix, called the Gaussian covariance matrix of μ , νμ is a measure on Rd 2 satisfying νμ ({0}) = 0 and Rd (|x| ∧ 1)νμ (dx) < ∞, called the Lévy measure of μ , and γμ is an element of Rd . The triplet (Aμ , νμ , γμ ) is uniquely determined by μ . Conversely, to any triplet (A, ν , γ ) there corresponds a unique μ ∈ ID such that A = Aμ , ν = νμ , and γ = γμ . Throughout this article Aμ , νμ , and γμ are used in this sense. A distribution μ on Rd is called selfdecomposable if, for any b > 1, there is a distribution μb such that (z) = μ (b−1 z)μ b (z), μ

z ∈ Rd .

(1.2)

Let L = L(Rd ) denote the class of selfdecomposable distributions on Rd . It is characterized in the following four ways. (a) A distribution μ on Rd is selfdecomposable if and only if μ ∈ ID and its Lévy measure νμ has a radial (or polar) decomposition

νμ (B) =

S

λ (d ξ )

∞ 0

1B (rξ )r−1 kξ (r)dr

(1.3)

for Borel sets B in Rd , where λ is a finite measure on the unit sphere S = {ξ ∈ Rd : |ξ | = 1} (if d = 1, then S is the two-point set {1, −1}) and kξ (r) is a nonnegative function measurable in ξ and decreasing and right-continuous in r. (See Proposition 3.1 for an exact formulation of the radial decomposition.)

Fractional Integrals and Extensions of Selfdecomposability

3

(b) Let {Zk : k = 1, 2, . . .} be independent random variables on Rd and Yn = ∑nk=1 Zk . Suppose that there are bn > 0 and γn ∈ Rd for n = 1, 2, . . . such that the law of bnYn + γn converges weakly to a distribution μ as n → ∞ and that {bn Zk : k = 1, . . . , n; n = 1, 2, . . .} is a null array (that is, for any ε > 0, max1≤k≤n P(|bn Zk | > ε ) → 0 as n → ∞). Then μ ∈ L. Conversely, any μ ∈ L is obtained in this way. (ρ ) (c) Given ρ ∈ ID, let {Xt : t ≥ 0} be a Lévy process on Rd (that is, a stochastic process continuous in probability, starting at 0, with time-homogeneous independent increments, with cadlag paths) having distribution ρ at time 1. If ∞− −s (ρ ) is |x|>1 log |x|ρ (dx) < ∞, then the improper stochastic integral 0 e dXs definable and its distribution ∞−

(ρ ) −s μ =L e dXs (1.4) 0

is selfdecomposable. Here L (Y ) denotes the distribution (law) of a random element Y . Conversely, any μ ∈ L is obtained in this way. On the other hand, (ρ ) if |x|>1 log |x|ρ (dx) = ∞, then 0∞− e−s dXs is not definable. (See Section 3.4 for improper stochastic integrals.) To see that μ of (1.4) is selfdecomposable, notice that ∞−

e 0

−s

(ρ ) dXs

=

log b

e 0

−s

(ρ ) dXs +

∞− log b

I1 and I2 are independent, and I2 =

∞−

e 0

− log b−s

(ρ ) dXlog b+s

=b

−1

(ρ )

e−s dXs

∞− 0

= I1 + I2,

(ρ )

e−s dYs ,

(ρ ) {Ys }

(ρ ) is identical in law with {Xs }, and hence μ satisfies (1.2). ≥ 0} be an additive process on Rd , that is, a stochastic process

where con(d) Let {Yt : t tinuous in probability with independent increments, with cadlag paths, and with Y0 = 0. If, for some H > 0, it is H-selfsimilar (that is, for any a > 0, the two processes {Yat : t ≥ 0} and {aH Yt : t ≥ 0} have an identical law), then the distribution μ of Y1 is in L. Conversely, for any μ ∈ L and H > 0, there is a process {Yt : t ≥ 0} satisfying these conditions and L (Y1 ) = μ . Historically, selfdecomposable distributions were introduced by Lévy [18] in 1936 and written in his 1937 book [19] under the name “lois-limites”, to characterize the limit distributions in (b). Lévy wrote in [18, 19] that this characterization problem had been posed by Khintchine, and Khintchine’s book [16] in 1938 called these distributions “of class L”. The book [9] of Gnedenko and Kolmogorov uses the same naming. Loève’s book [20] uses the name “selfdecomposable”. The property (c) gives a characterization of the stationary distribution of an Ornstein–Uhlenbeck type process (sometimes called an Ornstein–Uhlenbeck process driven by a Lévy process) {Vt : t ≥ 0} defined by Vt = e−t V0 +

t 0

(ρ )

es−t dXs ,

4

K. Sato (ρ )

where V0 and {Xt : t ≥ 0} are independent. The stationary Ornstein–Uhlenbeck type process and the selfsimilar process in the property (d) are connected via the so-called Lamperti transformation (see [11, 26]). For historical facts concerning (c) see [33], pp. 54–55. The proofs of (a)–(d) and many examples of selfdecomposable distributions are found in Sato’s book [39]. The main purpose of the present article is to give two families of subclasses of ID, with two continuous parameters, related to L, using improper stochastic integrals and extending the characterization (c) of L.

1.2 Nested Classes of Multiply Selfdecomposable Distributions If μ ∈ L, then, for any b > 1, the distribution μb in (1.2) is infinitely divisible and uniquely determined by μ and b. If μ ∈ L and μb ∈ L for all b > 1, then μ is called twice selfdecomposable. Let n be a positive integer ≥ 3. A distribution μ is called n times selfdecomposable, if μ ∈ L and if μb is n − 1 times selfdecomposable. Let L1,0 = L1,0 (Rd ) = L(Rd ) and let Ln,0 = Ln,0 (Rd ) be the class of n times selfdecomposable distributions on Rd . Then we have ID ⊃ L = L1,0 ⊃ L2,0 ⊃ L3,0 ⊃ · · · .

(1.5)

These classes and the class L∞ (Rd ) in Section 1.4 were introduced by Urbanik [52, 53] and studied by Sato [37] and others. (In [37, 52, 53] the class Ln,0 is written as Ln−1 , but this notation is inconvenient in this article.) An n times selfdecomposable distribution is characterized by the property that μ ∈ ID with Lévy measure νμ having radial decomposition (1.3) in (a) with kξ (r) = hξ (log r) for some function hξ (y) monotone of order n for each ξ (see Section 1.5 and Proposition 2.11 for the monotonicity of order n). In property (b), μ ∈ Ln,0 is characterized by the property that L (Zk ) ∈ Ln−1,0 for k = 1, 2, . . .. In (c), μ ∈ Ln,0 is characterized by ρ ∈ Ln−1,0 in (1.4). A direct generalization of (1.4) using exp(−s1/n ) or, equivalently, exp(−(n! s)1/n ) in place of e−s is also possible. In (d), μ ∈ Ln,0 if and only if, for any H, the corresponding process {Yt : t ≥ 0} satisfies L (Yt − Ys ) ∈ Ln−1,0 for 0 < s < t. The proofs are given in [12, 25, 33, 37].

1.3 Continuous-Parameter Extension of Multiple Selfdecomposability In 1980s Nguyen Van Thu [49–51] defined a continuous-parameter extension of Ln,0 , replacing the positive integer n by a real number p > 0. He introduced fractional times multiple Selfdecomposability and used fractional integrals and fractional difference quotients. On the one hand he extended the definition of n times


5

Selfdecomposability based on (1.2) to fractional times Selfdecomposability in the form of infinite products. On the other hand he extended essentially the formula (1.4) in the characterization (c), considering its Lévy measure. Directly using improper stochastic integrals with respect to Lévy processes, we will define and study the decreasing classes L p,0 for p > 0, which generalize the nested classes Ln,0 for n = 1, 2, . . .. Thus the results of Thu will be reformulated as a special case in a family L p,α with two continuous parameters 0 < p < ∞ and −∞ < α < 2. The definition of L p,α will be given in Section 1.6.

1.4 Stable Distributions and the Class L∞ Let μ be a distribution on Rd . Let 0 < α ≤ 2. We say that μ is strictly α -stable if (z)t = μ (t 1/α z), z ∈ Rd . We say that μ is α -stable if μ ∈ ID and, for any t > 0, μ (z)t = μ (t 1/α z) exp(iγt , z), μ ∈ ID and, for any t > 0, there is γt ∈ Rd such that μ d z ∈ R . (When μ is a δ -distribution, this terminology is not the same as in Sato [39].) Let S0α = S0α (Rd ) and Sα = Sα (Rd ) be the class of strictly α -stable distributions on Rd and the class of α -stable distributions on Rd , respectively. Let S = S(Rd ) d be the class of stable distributions on R . That is, S = 0 0 of a function f (s) on R in a suitable class is given by ∞

cp

r

(s − r) p−1 f (s)ds,

which is the interpolation (1 ≤ p < ∞) and extrapolation (0 < p ≤ 1) of the n times integration ∞ r

∞

dsn

sn

dsn−1 · · ·

∞ s2

f (s1 )ds1 =

1 (n − 1)!

∞ r

(s − r)n−1 f (s)ds.


7

However, we need to use fractional integrals of measures. Our definition is as follows. Let R+ = [0, ∞), R◦+ = (0, ∞) and B(E) denote the class of Borel sets in a space E. A measure σ is said to be locally finite on R [resp. R◦+ ] if σ ([a, b]) < ∞ for all a, b with −∞ < a 0. For a measure σ on R [resp. R◦+ ], let

σ (E) = c p

dr E

(r,∞)

(s − r) p−1σ (ds),

E ∈ B(R) [resp. B(R◦+ )].

(1.9)

Let D(I p ) [resp. D(I+p )] be the class of locally finite measures σ on R [resp. R◦+ ] such that σ is a locally finite measure on R [resp. R◦+ ]. Define I p σ (E) = σ (E),

E ∈ B(R)

[resp. I+p σ (E) = σ (E),

E ∈ B(R◦+ )]

for σ ∈ D(I p ) [resp. D(I+p )]. Thus I p and I+p are mappings from measures to measures on R and R◦+ , respectively. D(I p ) and D(I+p ) are their domains. We call a [0, ∞]-valued function f (r) on R [resp. R◦+ ] monotone of order p on R [resp. R◦+ ] if f (r) = c p

(r,∞)

(s − r) p−1σ (ds)

(1.10)

with some σ ∈ D(I p ) [resp. D(I+p )]. As will be shown in Example 2.17, functions monotone of order p ∈ (0, 1) have, in general, quite different properties from functions monotone of order p ∈ [1, ∞). We call f (r) completely monotone on R [resp. R◦+ ] if it is monotone of order p on R [resp. R◦+ ] for all p > 0. This definition of complete monotonicity differs from the usual one in that positive constant functions are not completely monotone. Typical completely monotone functions on R and R◦+ are e−r and r−α (α > 0), respectively. The properties of fractional integrals of functions are studied in M. Riesz [32], Ross (ed.) [35], Samko, Kilbas, and Marichev [36], Kamimura [15], and others. Williamson [56] studied fractional integrals of measures on R◦+ for p ≥ 1 and introduced the concept of p-times monotonicity. But we do not assume any knowledge of them. In Sections 2.1–2.3 we build the theory of the fractional integral mappings I p and p I+ for p ∈ (0, ∞) from the point of view that they are mappings from measures to q p measures. A basic relation is the semigroup property I q I p = I p+q and I+ I+ = I+p+q . p An important property that both I p and I+ are one-to-one is proved. The relation between the theories on R and R◦+ is not extension and restriction. We need both theories, as will be mentioned at the end of Section 6.2.

8

K. Sato

1.6 The Classes Kp,α and Lp,α Generated by Stochastic Integral Mappings The formula (1.4) gives a mapping Φ from ρ ∈ ID(Rd ) to μ ∈ ID(Rd ). Thus

Φρ = L

∞−

e

−s

0

(ρ ) dXs

.

(1.11)

The domain of Φ is the class of ρ for which the improper stochastic integral in (1.11) is definable. For functions f (s) in a suitable class, we are interested in the mapping Φ f from ρ ∈ ID to μ ∈ ID defined by

μ = Φf ρ = L

∞− 0

(ρ ) f (s)dXs

.

(1.12)

The domain D(Φ f ) is the class of ρ for which the improper stochastic integral in (1.12) is definable. The range is defined by R(Φ f ) = {Φ f ρ : ρ ∈ D(Φ f )}. Let us consider three families of functions. For 0 < p < ∞ and −∞ < α < ∞ let g¯ p,α (t) = c p j p,α (t) = c p gα (t) =

1 t

1

∞

t

(1 − u) p−1u−α −1 du,

0 < t ≤ 1,

(− log u) p−1 u−α −1 du,

u−α −1 e−u du,

t

0 < t ≤ 1,

0 < t < ∞,

(1.13) (1.14) (1.15)

and a¯ p,α = g¯ p,α (0+), b p,α = j p,α (0+), aα = gα (0+). If α < 0, then a¯ p,α = Γ−α /Γp−α , b p,α = (−α )−p , and aα = Γ−α . If α ≥ 0, then a¯ p,α = b p,α = aα = ∞. Let t = f¯p,α (s), l p,α (s), and fα (s) be the inverse functions of s = g¯ p,α (t), j p,α (t), and gα (t), respectively. When α < 0, extend f¯p,α (s) for s ≥ a¯ p,α , l p,α (s) for s ≥ b p,α , and fα (s) for s ≥ aα to be zero. Define

Φ¯ p,α = Φ f¯p,α ,

Λ p,α = Φl p,α ,

Ψα = Φ fα .

Sato [42] studied the mapping Ψα and the mapping Φβ ,α = Φ fβ ,α , −∞ < β < α < ∞, for the inverse function fβ ,α (s) of the function gβ ,α (t) defined by gβ ,α (t) = cα −β

1 t

(1 − u)α −β −1u−α −1du,

0 < t ≤ 1.

To make the parametrization more convenient, we use Φ¯ p,α = Φα −p,α . For Φ¯ p,α , Λ p,α , and Ψα , the domains will be characterized. In the analysis of the domains, asymptotic behaviors of f¯p,α (s), l p,α (s), and fα (s) for s → ∞ are essential.


9

The behaviors of f¯p,α (s) and fα (s) are similar, but the behavior of l p,α (s) is different from them. If α ≥ 2, then D(Φ¯ p,α ) = D(Λ p,α ) = D(Ψα ) = {δ0 }. So we will only consider −∞ < α < 2. Define K p,α = K p,α (Rd ) = R(Φ¯ p,α ),

(1.16)

L p,α = L p,α (R ) = R(Λ p,α ).

(1.17)

d

It is clear that g¯1,α (t) = j1,α (t), and hence

Φ¯ 1,α = Λ1,α ,

K1,α = L1,α

for −∞ < α < 2.

(1.18)

Since g¯1,0 (t) = j1,0 (t) = − logt, 0 < t ≤ 1, and f¯1,0 (s) = l1,0 (s) = e−s , s ≥ 0, we have Φ¯ 1,0 = Λ1,0 = Φ , K1,0 = L1,0 = L (1.19) So K p,α and L p,α give extensions, with two continuous parameters, of the class L of selfdecomposable distributions. Since l p,0 (s) = exp(−(Γp+1 s)1/p ), s ≥ 0, the class L p,0 coincides with the class of n times selfdecomposable distributions if p is an integer n. The following are some of the new results in this article. For any α and p with −∞ < α < 2 and p > 0, any μ ∈ K p,α has Lévy measure νμ having a radial decomposition

νμ (B) =

S

λ (d ξ )

∞ 0

1B (rξ )r−α −1 kξ (r)dr

(1.20)

with kξ (r) measurable in (ξ , r) and monotone of order p on R◦+ in r, and any μ ∈ L p,α has Lévy measure νμ having a radial decomposition

νμ (B) =

S

λ (d ξ )

∞ 0

1B (rξ )r−α −1 hξ (log r)dr

(1.21)

with hξ (y) measurable in (ξ , y) and monotone of order p on R in y. If −∞ < α < 1, < 2, then this propthen this property of νμ characterizes K p,α and L p,α . If 1 < α erty of ν combined with the property of mean 0 (that is, μ Rd |x| μ (dx) < ∞ and x μ (dx) = 0) characterizes K and L . We will introduce the notion of weak p,α p,α Rd mean of infinitely divisible distributions in Section 3.3. If α = 1, then the property above of νμ and the property of weak mean 0 characterize K p,1 ; the case of L p,1 is still open. For each fixed α , the classes K p,α and L p,α are strictly decreasing as p increases and at the limit there appear connections with R(Ψα ) and with the class L∞ of completely selfdecomposable distributions. Namely, define K∞,α =

0 0 on R, then the restriction of f to R◦+ is monotone of order p on R◦+ . (ii) If f is monotone of order p ≥ 1, then f is finite-valued and decreasing. For p = 1 this is obvious. For p > 1 this follows from Corollary 2.6 to be given later. (iii) If f is monotone of order p ∈ (0, 1), then f is finite almost everywhere, but f possibly takes the infinite value at some point and f is not necessarily decreasing. See Example 2.17 (a), (b), and (d). Proposition 2.1. Let p > 0. It holds that D(I p ) = M∞p−1 (R), D(I+p )

(2.1)

= M∞p−1 (R◦+ ).

(2.2)

Proof. Let σ be a locally finite measure on R [resp. R◦+ ]. Let −∞ < a < b < ∞

of (1.9) satisfies [resp. 0 < a < b < ∞]. Then σ

σ ([a, b]) = c p = cp

b

dr a

(a,∞)

= c p+1

(r,∞)

(s − r) p−1 σ (ds)

σ (ds)

(b,∞)

b∧s a

(s − r) p−1dr

((s − a) p − (s − b) p)σ (ds) + c p+1

which is finite if and only if

(1,∞) s

p−1 σ (ds)

(a,b]

(s − a) pσ (ds),

< ∞, since

(s − a) p − (s − b) p = s p ((1 − a/s) p − (1 − b/s) p) ∼ p(b − a)s p−1 as s → ∞.

q

Corollary 2.2. If 0 < q < p, then D(I p ) ⊂ D(I q ) and D(I+p ) ⊂ D(I+ ). Proposition 2.3. Let p > 0. Let α > −1 and β > 0. (i) Let σ ∈ D(I p ) [resp. D(I+ )]. Then I p σ ∈ Mα∞ (R) [resp. I+ σ ∈ Mα∞ (R◦+ )] if and only if σ ∈ M∞p+α (R) [resp. M∞p+α (R◦+ )]. (ii) Let σ ∈ D(I+p ). Then I+p σ ∈ Mα0 (R◦+ ) if and only if σ ∈ M0p+α (R◦+ ). (iii) Let σ ∈ D(I p ). Then (−∞,0) eβ r (I p σ )(dr) < ∞ if and only if (−∞,0) eβ s σ (ds) < ∞. p

p

Assertion (i) is the right-tail fattening property of I p [resp. tail fattening property of I+p ]. Assertion (ii) is the head thinning property of I+p .


13

Proof. To see (i), let σ = I p σ [resp. I+p σ ]. We have ∞ 1

rα σ (dr) = c p = cp = cp

∞

rα dr

(1,∞)

(1,∞)

σ (ds)

rα (I+p σ )(dr) = c p = cp = cp

(s − r) p−1 σ (ds)

s 1

rα (s − r) p−1 dr

s p+α σ (ds)

0

(r,∞)

1

(dr) < ∞ if and only if Hence (1,∞) rα σ Proof of (ii) is as follows. We have 1

1

(1,∞) s

rα dr

(r,∞)

(0,∞)

(0,1]

σ (ds)

1/s

uα (1 − u) p−1du.

p+α σ (ds)

0

1

< ∞.

(s − r) p−1σ (ds)

1∧s 0


f (s)σ (ds) + c p

(1,∞)

g(s)σ (ds)

where f (s) = g(s) =

s 0

1 0

rα (s − r) p−1dr

for 0 < s ≤ 1,


for s > 1.

Since f (s) = sα +p and g(s) = sα +p

1/s 0

1 0

uα (1 − u) p−1du

uα (1 − u) p−1du ∼ (α + 1)−1s p−1 ,

s → ∞,

and since (1,∞) s p−1 σ (ds) < ∞, we obtain the assertion. Let us prove (iii). We have 0 −∞

eβ r (I p σ )(dr) = c p = cp = cp

0

−∞

R

eβ r dr

σ (ds)

(−∞,0]

(r,∞) s∧0 −∞

(s − r) p−1 σ (ds)

eβ r (s − r) p−1dr

f (s)σ (ds) + c p

(0,∞)

g(s)σ (ds),

14

K. Sato

where f (s) = g(s) = Notice that

g(s) = e

(1,∞) s

−∞ 0 −∞


for s ≤ 0,


for s > 0.

f (s) = eβ s

and

Using

s

p−1 σ (ds)

βs

∞ s

∞

e−β uu p−1 du

0

e−β uu p−1 du ∼ β −1 s p−1 ,

s → ∞.

< ∞, we can show the result.

Proposition 2.4. For any p > 0 and q > 0, I q I p = I p+q

q

p+q

and I+ I+p = I+ .

(2.3)

As always an equality of mappings includes the assertion that the domains of both hands are equal.

= I p σ [resp. Lemma 2.5. Let p > 0 and q > 0. If σ ∈ D(I p ) [resp. D(I+p )] and σ p I+ σ ], then

cq

(u,∞)

(r − u)q−1 σ (dr) = c p+q

(u,∞)

(s − u) p+q−1σ (ds)

(2.4)

for u ∈ R [resp. R◦+ ]. Proof. We have (u,∞)

(dr) = cq (r − u)q−1 σ

∞ u

cq (r − u)q−1 dr

= c p cq = c p cq

= c p+q which is (2.4).

σ (ds)

(u,∞)

(u,∞)

s u

(r,∞)

c p (s − r) p−1σ (ds)

(r − u)q−1 (s − r) p−1 dr

(s − u) p+q−1σ (ds)

1 0

(1 − v)q−1v p−1 dv

(by change of variables v = (s − r)/(s − u))

(u,∞)

(s − u) p+q−1σ (ds),

Proof of Proposition 2.4. We prove the first equation in (2.3), but the proof of the second one is formally the same. The domain of I q I p is defined to be {σ ∈ D(I p ) : I p σ ∈ D(I q )}. It follows from Propositions 2.1 and 2.3 (i) that


σ ∈ D(I q I p )

⇔

σ ∈ M∞p−1 (R),

⇔

σ ∈ M∞p+q−1(R)

⇔

σ ∈ D(I p+q ).

15

I p σ ∈ Mq−1 ∞ (R)

If σ ∈ M∞p+q−1 (R), then Lemma 2.5 shows that (I q (I p σ ))(du) = (I p+q σ )(du). Corollary 2.6. Let 0 < q < p. If a function f is monotone of order p on R [resp. R◦+ ], then f is monotone of order q on R [resp. R◦+ ].

2.2 One-to-One Property We will prove an important result that I p and I+p are one-to-one. We prepare auxiliary mappings Dq and Dq+ and two lemmas, suggested by Kamimura [15]. q

Definition 2.7. Let 0 < q < 1. Let D(Dq ) [resp. D(D+ )] be the class of locally finite measures ρ on R [resp. R◦+ ] absolutely continuous with density g(s) such that ∞ r

(s − r)−q−1 |g(s) − g(r)|ds < ∞

for a. e. r ∈ R [resp. R◦+ ]

(2.5)

and that the signed measure ρ defined by

∞ −q−1 ρ (dr) = qc1−q (s − r) (g(s) − g(r))ds dr

(2.6)

r

has locally finite variation on R [resp. R◦+ ]. Define Dq ρ = ρ

[resp. D+ ρ = ρ ] q

(2.7)

for ρ ∈ D(Dq ) [resp. D(Dq+ )]. q

The reason for introducing the mappings Dq and D+ is seen from the following lemma. Lemma 2.8. Let 0 < q < p < 1 and let σ ∈ D(I p ) [resp. D(I+p )]. Then I p σ ∈ D(Dq ) p )] and [resp. I+p σ ∈ D(D+ (Dq I p σ )(dr) = q

Γp−q (qC p,q − 1) (I p−qσ )(dr) ΓpΓ1−q p−q

[resp. the same equality with D+ , I+p , and I+ C p,q =

1 0

(2.8)

in place of Dq , I p , and I p−q ], where

(1 − u)−q−1(u p−1 − 1)du.

(2.9)

16

K. Sato

Proof. Let ρ = I p σ [resp. I+p σ ]. Then ρ (ds) = g(s)ds with g(s) = c p s) p−1 σ (du). For s > r we have g(s) − g(r) = −c p = −c p

(r,s] (r,s]

(u − r)

p−1

σ (du) + c p

(s,∞)

(s,∞) (u −

((u − s) p−1 − (u − r) p−1)σ (du)

(u − r) p−1σ (du) + (1 − p)c p

(s,∞)

σ (du)

s r

(u − v) p−2dv.

Let J1 =

∞ r

(s − r)−q−1 ds

J2 = (1 − p) ∞

Then

r

∞ r

(r,s]

(u − r) p−1σ (du),

−q−1

(s − r)

ds

σ (du)

(s,∞)

s r

(u − v) p−2dv.

(s − r)−q−1 |g(s) − g(r)|ds ≤ c p (J1 + J2 ).

Since σ ∈ D(I p−q ) [resp. D(I+p−q )], we have J1 =

(r,∞)

= q−1

(u − r) p−1σ (du)

(r,∞)

J2 = (1 − p) = (1 − p) = (1 − p) = (1 − p) = (1 − p) = (1 − p) = C p,q

∞ u

(s − r)−q−1 ds

(u − r) p−q−1σ (du) < ∞ for a. e. r ∈ R [resp. R◦+ ],

(r,∞)

(r,∞)

(r,∞)

(r,∞)

(r,∞)

(r,∞)

(r,∞)

σ (du) σ (du) σ (du) σ (du)

u r

u r

(u − v) p−2dv (u − v) p−1dv

1

u

dt 0

1 0

r

t −p dt

u v

1 0

(s − r)−q−1ds (u − r − t(u − v))−q−1dt

(u − v) p−1(u − r − t(u − v))−q−1dv t(u−r) 0

(u − r) p−q−1σ (du) (u − r) p−q−1σ (du)

1 0

1 0

w p−1 (u − r − w)−q−1dw

t −p dt

t 0

x p−1 (1 − x)−q−1dx

x p−1 (1 − x)−q−1dx

1

t −p dt

x

(u − r) p−q−1σ (du) < ∞ for a. e. r ∈ R [resp. R◦+ ],

where C p,q =

1 0

x p−1 (1 − x)−q−1(1 − x1−p)dx = C p,q


17

and the finiteness of C p,q is clear since (1 − x)−q−1(1 − x1−p) ∼ (1 − p)(1 − x)−q as x ↑ 1. We have thus shown (2.5) and ∞ r

(s − r)−q−1 (g(s) − g(r))ds = c p (J2 − J1 ) = c p (C p,q − q−1)

(r,∞)

(u − r) p−q−1σ (du).

Hence I p σ ∈ D(Dq ) [resp. I+p σ ∈ D(Dq+ )] and (D p I p σ )(dr) = c1−q c p (qC p,q − 1)

(u − r) p−q−1σ (du) dr

(r,∞) p−q

= Γp−qc1−q c p (qC p,q − 1)I

σ (dr)

on R, and similarly on R◦+ .

Lemma 2.9. Let p > 0 and let σ I qσ → σ

∈ D(I p )

[resp.

D(I+p )].

q [resp. I+ σ →σ

vaguely on R

Then,

vaguely on R◦+ ]

(2.10)

as q ↓ 0, that is, for all continuous functions f with compact support in R [resp. R◦+ ],

f (s)I q σ (ds) →

f (s)σ (ds)

[resp.

f (s)I+ σ (ds) → q

f (s)σ (ds)]

(2.11)

as q ↓ 0. Proof. We give the proof in the case R, but the case R◦+ is similar. First recall that σ ∈ D(I p ) implies σ ∈ D(I q ) for 0 < q ≤ p. Assume that f is nonnegative, continuous with support in [a, b] for some a < b. It is enough to show (2.11) for such f . Notice that R

f (s)I q σ (ds) =

R

f (r)dr

where gq (s) =

(r,∞)

s −∞

cq (s − r)q−1 σ (ds) =

R

gq (s)σ (ds),

cq (s − r)q−1 f (r)dr.

We claim that gq (s) → f (s),

q↓0

(2.12)

for s ∈ R. We have gq (s) = 0 = f (s) for s ≤ a. Fix s > a. Let q be such that a < s − q < s. Then, as q ↓ 0, |gq (s) − f (s)| ≤ cq

s−q

s

(s − r)q−1 f (r)dr + cq (s − r)q−1 | f (r) − f (s)|dr a s−q s q−1 + cq (s − r) dr − 1 f (s) s−q

18

K. Sato

= J1 + J2 + J3 , J1 ≤ cq || f ||

s−q a

(s − r)q−1 dr = cq+1 || f ||((s − a)q − qq) → 0,

where || f || = maxs∈R f (s), J2 ≤ max | f (r) − f (s)|cq+1 qq → 0, r∈[s−q,s] J3 = |cq+1 qq − 1|

f (s) → 0.

This proves (2.12). If s > a, then gq (s) ≤ cq || f ||

s a

(s − r)q−1 dr = cq+1 || f ||(s − a)q ≤ const ((s − a) ∨ 1) p

for 0 < q ≤ p. If s > b + 1, then gq (s) ≤ cq || f ||

b a

(s − r)q−1 dr ≤ cq || f ||(b − a)(s − b)q−1 ≤ const (s − b) p−1

for 0 < q ≤ p. Now, since σ ∈ M∞p−1 (R), we can use the dominated convergence theorem and obtain R

gq (s)σ (ds) →

R

f (s)σ (ds),

q ↓ 0,

completing the proof. Theorem 2.10. For any p > 0, I p and I+p are one-to-one.

Proof. Assume that p < 1. Suppose that σ1 , σ2 ∈ D(I p ) satisfy I p σ1 = I p σ2 . Let 0 < q < p. By virtue of Lemma 2.8, I p σ j ∈ D(Dq ) for j = 1, 2 and (2.8) holds for σ = σ1 , σ2 . We have Dq I p σ1 = Dq I p σ2 . If qC p,q − 1 = 0, then it follows that I p−q σ1 = I p−q σ2 . From the definition (2.9), C p,q is positive and strictly increasing with respect to q. Hence, either qC p,q − 1 = 0 for all q ∈ (0, p) or there is q0 ∈ (0, p) such that qC p,q − 1 = 0 for all q ∈ (0, p) \ {q0}. Thus qC p,q − 1 = 0 Hence

I p−q σ1 = I p−q σ2

for all q ∈ (0, p) sufficiently close to p.

(2.13)

for all q ∈ (0, p) sufficiently close to p.

Now, letting q ↑ p and using Lemma 2.9, we obtain σ1 = σ2 . It follows that I p is one-to-one for 0 0. The proof for I+p is similar.


19

2.3 More Properties and Examples When p is a positive integer, we have the following characterization. This is a result of Williamson [56]. It is given also in Lemmas 3.2 and 3.4 of Sato [37] based on Widder’s book [55]. Proposition 2.11. (i) A function f (r) on R [resp. R◦+ ] is monotone of order 1 if and only if it is decreasing and right-continuous on R [resp. R◦+ ] and tends to 0 as r → ∞. (ii) Let n be an integer ≥ 2. A function f on R [resp. R◦+ ] is monotone of order n if and only if ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f (r) tends to 0 as r → ∞ and is n − 2 times differentiable on R [resp. R◦+ ] with (−1) j f ( j) ≥ 0 for j = 0, 1, . . . , n − 2, and with

(−1)n−2 f (n−2)

(2.14)

being decreasing and convex.

Corollary 2.12. Let n be an integer ≥ 1. Suppose that f is n times differentiable on R [resp. R◦+ ]. Then f is monotone of order n if and only if (−1) j f ( j) ≥ 0 on R [resp. R◦+ ] for j = 0, 1, . . . , n, and f (r) → 0 as r → ∞. Thus the concept of complete monotonicity of f on R◦+ coincides with that in Widder [55] and Feller [8] except the condition that limr→∞ f (r) = 0. An integral representation of a completely monotone function on R◦+ (as the Laplace transform of a measure on R◦+ ) is obtained from Bernstein’s theorem. A completely monotone function on R is also represented by the Laplace transform of a measure on R◦+ . Proof of Proposition 2.11. In this proof we consider the case R. In the case R◦+ , replace R by R◦+ . f is monotone of order 1 on R if and only if f (r) = Let us prove (i). Recall that 0 (R), hence if and only if f (r) is finite, decreasing, and σ (ds) for some σ ∈ M ∞ (r,∞) right-continuous on R and tends to 0 as r → ∞. Proof of (ii) is as follows. Let n ≥ 2. A function f is monotone of order n on R if and only if, for some σ ∈ Mn−1 ∞ (R),

1 (s − r)n−1 σ (ds) = σ (ds) (n − 1)! (r,∞) (r,∞) 1 = (s − u)n−2σ (ds). du (r,∞) (u,∞) (n − 2)!

f (r) =

s r

1 (s − u)n−2du (n − 2)!

If f is monotone of order n on R, then f (r) → 0 as r → ∞, since f is monotone of order 1 on R. If f is monotone of order 2 on R, then f (r) =

(r,∞)

σ ((u, ∞))du

(2.15)

20

K. Sato

and hence f is decreasing and convex. Conversely, if f (r) is decreasing, convex, and convergent to 0 as r → ∞, then f is written as in (2.15) with some σ ∈ M1∞ (R) and hence f is monotone of order 2 on R. Now let n ≥ 3 and suppose that assertion n − 1 in place of n. If f is (ii) is1 true withn−2 (s − u) σ (ds) is monotone of monotone of order n on R, then g(u) = (u,∞) (n−2)! order n − 1 on R and, a fortiori, continuous and hence − f (r) = g(r), which shows that (2.14) is satisfied. Conversely, suppose that f satisfies (2.14). Then − f (r) → 0 as r → ∞, since otherwise f (r) goes to −∞ as r → ∞. Hence (2.14) is satisfied with − f in place of f and with n − 1 in place of n. Hence − f (u) =

1 (s − u)n−2σ (ds) (n − 2)! (u,∞)

for some σ ∈ Mn−2 ∞ (R). Since − f (u) is continuous and since f (r) → 0 as r → ∞, ∞ we have f (r) = r f (u)du and hence

f (r) =

1 (s − r)n−1σ (ds). (n − 1)! (r,∞)

As (2.14) implies that f is locally integrable on R, σ belongs to D(I n ) and f is monotone of order n on R. Let us give some necessary conditions for f to be monotone of order p. Proposition 2.13. Suppose that f is monotone of order p on R [resp. R◦+ ] for some p > 0 and that f is not identically zero. Then: (i) f is lower semi-continuous on R [resp. R◦+ ]. (ii) Either f (r) > 0 for all r ∈ R [resp. R◦+ ] or there is a ∈ R [resp. R◦+ ] such that f (r) > 0 for r < a and f (r) = 0 for r ≥ a. (iii) In the case of R, lim inf ( f (r)/|r| p−1 ) > 0. r→−∞

(iv) In the case of R◦+ , lim inf f (r) > 0. r↓0

Proof. The function f satisfies (1.10) for some σ ∈ M∞p−1 (R) [resp. M∞p−1 (R◦+ )] with σ = 0. To show (i), use Fatous lemma and see lim inf f (r ) ≥ c p r →r

= cp

lim inf(1(r ,∞) (s)(s − r ) p−1 )σ (ds) r →r

(r,∞)

(s − r) p−1 σ (ds) = f (r),

that is, f is lower semi-continuous.To see (ii), note that if f (r0 ) > 0 for some r0 , then f (r) > 0 for all r ≤ r0 , because (r0 ,∞) (s− r0 ) p−1 σ (ds) > 0 shows that there is a point s0 in the support of σ such that s0 > r0 . To prove (iii), choose −∞ < a 0. Let r < a. Then


f (r) ≥ c p

(a,b)

(s − r) p−1σ (ds) ≥

21

c p (b − r) p−1σ ((a, b)) if p ≤ 1, c p (a − r) p−1σ ((a, b)) if p > 1.

Hence the assertion follows. Assertion (iv) is proved similarly to (iii).

Proposition 2.14. Suppose that f is monotone of order p on R [resp. R◦+ ] for some p > 1. Then f is absolutely continuous on R [resp. R◦+ ]. Proof. Consider the case of R. We have (1.10) for f with some σ ∈ D(I p ). Since I p = I 1 I p−1 , it follows from Lemma 2.5 that f (r) =

(r,∞)

(I p−1 σ )(ds) =

∞

g(s)ds r

for some g(s) ≥ 0. The case of R◦+ is similar.

Let S = Sd−1 = {ξ ∈ Rd : |ξ | = 1}. This is the (d − 1)-dimensional unit sphere in if d ≥ 2 and the two-point set {−1, 1} if d = 1. A family {σξ : ξ ∈ S} of locally finite measures on R [resp. R◦+ ] is called a measurable family if σξ (E) is measurable in ξ ∈ S for every E ∈ B(R) [resp. B(R◦+ )]. If {σξ : ξ ∈ S} is a measurable family, then, (a) for any [0, ∞]-valued function f (r, s) measurable in (r, s), f (r, s)σξ (ds) is measurable in (ξ , r), and (b) for any a > 0, σξ ((r, r + a]) is measurable in (ξ , r). To see (a), use the monotone class theorem. To see (b), apply (a) to f (r, s) = 1(r,r+a] (s).

Rd

Proposition 2.15. Let p > 0. If {σξ : ξ ∈ S} is a measurable family of measures in M∞p−1 (R) [resp. M∞p−1 (R◦+ )], then {I p (σξ ) : ξ ∈ S} [resp. {I+p (σξ ) : ξ ∈ S}] is a measurable family. Proof. Notice that, for any E ∈ B(R) I p (σξ )(E) = =

dr

E

R

(r,∞)

σξ (ds)

c p (s − r) p−1σξ (ds)

E∩(−∞,s)

c p (s − r) p−1dr,

which is measurable in ξ . The case of R◦+ is similar.

Proposition 2.16. Let p > 0 and let {σξ : ξ ∈ S} ⊂ M∞p−1 (R) [resp. M∞p−1 (R◦+ )]. If {I p (σξ ) : ξ ∈ S} [resp. {I+p (σξ ) : ξ ∈ S}] is a measurable family, then {σξ : ξ ∈ S} is a measurable family. Proof. Consider the case of R. The case of R◦+ is similar. Let {I p (σξ )} be a measurable family. For each ξ I p (σξ )(E) =

E

gξ (r)dr,

gξ (r) =

(r,∞)

c p (s − r) p−1σξ (ds).

22

K. Sato

Let g ξ (r) = lim inf n n→∞

r+1/n r

gξ (r )dr = lim inf nI p (σξ )((r, r + 1/n]). n→∞

Then g ξ (r) is measurable in (ξ , r) and, by Lebesgue’s differentiation theorem, gξ (r) = g ξ (r) for a. e. r for every fixed ξ . Thus I p (σξ )(dr) = g ξ (r)dr. Suppose 0 < p < 1. Let 0 < q < p. Then {Dq I p (σξ ) : ξ ∈ S} is a measurable family. It follows from Lemma 2.8 and (2.13) that {I p−q(σξ ) : ξ ∈ S} is a measurable family for q sufficiently close to p. Hence, by Lemma 2.9, {σξ : ξ ∈ S} is a measurable family. Now, for any p > 0, write p = np with positive integer n and 0 0. In the following, σ is in M∞p−1 (R) or in M∞p−1 (R◦+ ) and we write (s − r) p−1 σ (ds) (2.16) f p (r) = c p (r,∞)

for r ∈ R or for r

∈ R◦+ .

Thus f p is monotone of order p on R or on R◦+ .

(a) A δ -distribution located at x is denoted by δx . Let σ = δa with a ∈ R [resp. R◦+ ]. Then c p (a − r) p−1, r < a, f p (r) = 0, r ≥ a. Hence f p (r) is strictly increasing for r < a if p < 1; f p equals 1 for r < a if p = 1; f p is not continuous if p ≤ 1. If p > 1, then f p is strictly decreasing for r < a and continuous on R [resp. R◦+ ]. For any p > p, f p is not monotone of order p . Indeed, otherwise Proposition 2.4 and Theorem 2.10 show that δa = I p −p τ [resp. I+p −p τ ] for some τ ∈ M∞p −1 (R) [resp. M∞p −1 (R◦+ )], which is absurd since

I p −p τ [resp. I+p −p τ ] is absolutely continuous. Notice that this function f p (r) has the following property: if α ∈ R satisfies α (p − 1) > −1, then f p (r)α is monotone of order α (p − 1) + 1 and not monotone of order p for any p > α (p − 1) + 1. (b) Let −∞ < a < b < ∞ [resp. 0 < a < b < ∞] and let σ (ds) = 1(a,b] (s)ds. Then

f p (r) =

Thus

⎧ p p ⎪ ⎪ ⎨c p+1 ((b − r) − (a − r) ),

r < a,

c p+1 ⎪ ⎪ ⎩0,

r ≥ b.

(b − r) p,

a ≤ r < b,

f p (r) = c p ((a − r) p−1 − (b − r) p−1) for r < a.

Hence, if p < 1, then f p is strictly increasing for r ≤ a and strictly decreasing for a ≤ r ≤ b. For all p > 0, f p is continuous on R [resp. R◦+ ]. For any p > p, f p is not monotone of order p on R [resp. R◦+ ]. Indeed, otherwise σ = I p −p τ

[resp. I+p −p τ ] for some τ ∈ M∞p −1 (R) [resp. M∞p −1 (R◦+ )], which contradicts Proposition 2.13.


23

(c) Let σ (ds) = s−α ds on R◦+ with α > p. Then σ ∈ M∞p−1 (R◦+ ) and the function f p is monotone of order p on R◦+ and ∞

f p (r) = c p

r

(s − r) p−1s−α ds = c p r p−α

for r > 0, where c+ = c p

∞ 0

∞ 1

(u − 1) p−1u−α du = c+ r p−α

u p−1(u + 1)−α du = c p B(p, α − p) = Γα −p /Γα .

(d) Suppose 0 α > p and b ∈ R. Then σ ∈ M∞p−1 (R) and f p is monotone of order p on R and ⎧ p−α , ⎪ r < b, ⎪ ⎨c− (b − r) f p (r) = ∞, r = b, ⎪ ⎪ ⎩c (r − b) p−α , r > b, +

where c+ is the same as in (c) and c− = c p

∞ 0

(u + 1) p−1u−α du = c p B(1 − α , α − p) = Γ1−α Γα −p /(ΓpΓ1−p ).

Note that f p (b) = c p b∞ (s − b) p−α −1 ds = ∞. This f p is a (0, ∞]-valued continuous function on R, strictly increasing on (−∞, b), equal to ∞ at b, and strictly decreasing on (b, ∞). For any p > p, this f p is not monotone of order p by the same reason as in (b). (e) Let 0 0, n = 1, 2, . . ., satisfying

∑

Cn +

bn ∈B∩(−∞,1]

Let

σ (ds) =

∑

bn ∈B∩(1,∞)

Cn bnp−α < ∞.

∞

∑ Cn (s − bn)−α 1(bn ,∞)(s)ds.

n=1

Then σ ∈ M∞p−1 (R), since we have (1,∞)

s p−1 σ (ds) =

∞

∑ Cn

n=1

∞ bn ∨1

s p−1 (s − bn)−α ds < ∞,

noting that, for bn ≤ 1, ∞ 1

s p−1 (s − bn)−α ds ≤

∞ 1

s p−1 (s − 1)−α ds =

= B(1 − α , α − p)

∞ 0

(u + 1) p−1u−α du

24

K. Sato

and, for bn > 1, ∞ bn

s p−1 (s − bn )−α ds = bnp−α

∞ 1

u p−1(u − 1)−α du = bnp−α B(1 − α , α − p).

Let f p,α ,b (s) denote the function in (d). Then f p (r) =

∞

∑ Cn f p,α ,bn (r)

n=1

and f p (bn ) = ∞ for n = 1, 2, . . .. If the set B has supremum ∞, then lim sup f p (s) = ∞. If B is a dense set in R, then f p (s) is finite almost everywhere s→∞

but infinite on the dense set. Example 2.18. (a) Let f (r) = r−β for r > 0 with β > 0. Then f is completely monotone on R◦+ , because, for any p > 0, we can choose α = p + β and apply Example 2.17 (c). Alternatively, use Proposition 2.11. monotone on R. Use (b) Let f (r) = e−r for r ∈ R. Then f is completely Proposition 2.11 or c p r∞ (s − r) p−1e−s ds = c p 0∞ u p−1e−u−r du = e−r . (c) Let arcsin(1 − r), 0 < r < 1, f (r) = 0, r ≥ 1. Then f is monotone of order 2 on R◦+ , since it is decreasing and convex. For any p > 2, f is not monotone of order p on R◦+ . To prove this, suppose f is monotone of order p > 2 on R◦+ . Then f (r)dr = I+p σ for some σ ∈ M∞p−1 (R◦+ ). 1 τ with τ = I p−1 σ . On the other hand Hence f (r)dr = I+ + f (r) =

∞

g(s)ds r

with g(s) = (1 − (1 − s)2)−1/2 1(0,1) (s).

Hence τ (ds) = g(s)ds by Theorem 2.10. Hence g(s) is equal almost everywhere on R◦+ to a function monotone of order p − 1. Since p − 1 > 1, it follows that g(s) is equal almost everywhere on R◦+ to an absolutely continuous function (Proposition 2.14). This is absurd. (d) Let − log r, 0 < r < 1, f (r) = 0, r ≥ 1. Then, similarly to the previous example, f is monotone of order 2 on R◦+ but is not monotone of order p on R◦+ for any p > 2. √ Example 2.19. Let g(r) = r2 + 1 − r, r ∈ R, and hα (r) = g(r)α , r ∈ R, with α ∈ (0, ∞). The function g is monotone of order 2 on R, since g(r) > 0, −g (r) = 1 − r(r2 + 1)−1/2 > 0, and


25

g (r) = (r2 + 1)−1/2 − r2 (r2 + 1)−3/2 = (r2 + 1)−3/2 > 0, g(r) = |r| 1 + |r|−2 − r = |r|(1 + O(|r|−2 )) − r = O(r−1 ), r → ∞. Let us show the following. (a) For every α > 0, hα is not monotone of order p on R for any p > α + 1. (b) For every α > 0, hα is monotone of order 1 on R. (c) The following statement is true for n = 1, 2, 3. For any α ≥ n, hα is monotone of order n + 1 on R. We have g(r) = 2|r| + O(|r|−1 ), r → −∞. Hence we see (a) by virtue of Proposition 2.13 (iii), because hα (r)/|r| p−1 ∼ 2α /|r| p−α −1 as r → −∞. We have (b), since √ α ( r2 + 1 − r)α −1 −α hα √ (r − r2 + 1) = √ , (2.17) hα = r2 + 1 r2 + 1 which is negative on R. We have (c) for n = 1, since hα

=α

rhα hα √ − (r2 + 1)3/2 r2 + 1

=

α hα (r + α r2 + 1), (r2 + 1)3/2

(2.18)

which is positive on R for α ≥ 1. The following recursion formula is known for the derivatives of hα ([30] p. 41): ( j+2)

(r2 + 1)hα

( j+1)

+ (2 j + 1)rhα

( j)

+ ( j2 − α 2 )hα = 0.

(2.19)

Indeed, this is true for j = 0 from (2.17) and (2.18); if (2.19) is true for a given j ≥ 0, then its differentiation shows that it is true with j + 1 in place of j. Now let us prove (c) for n = 2. It follows from (2.17), (2.18), and (2.19) that 2 (r2 + 1)h α = −3rhα − (1 − α )hα =

−3α rhα (1 − α 2 )α hα 2 + 1) + √ (r + α r (r2 + 1)3/2 r2 + 1

−α hα (3 α r r2 + 1 + (α 2 + 2)r2 + (α 2 − 1)) (r2 + 1)3/2 −α hα 3 2 [ α ( r + 1 + r)2 +(α − 2)(α − 1)r2 + (α − 2)(α + 12 )], = 2 (r + 1)3/2 2 =

which is negative on R for α ≥ 2. Let us prove (c) for n = 3. We have (4)

2 (r2 + 1)hα = −5rh α − (4 − α )hα 5α rhα = 2 (3α r r2 + 1 + (α 2 + 2)r2 + (α 2 − 1)) (r + 1)5/2 α hα (r + α r2 + 1) − (4 − α 2) 2 (r + 1)3/2

26

K. Sato

α hα 2 2 2 + 1 + (α 2 − 4)α r2 + 1 = 2 [( α + 11) α r r (r + 1)5/2 + 6(α 2 + 1)r3 + 3(2α 2 − 3)r] α hα 2 3 [ ( α + 1)( r2 + 1 + r)3 + 32 (α 2 − 9)r = 2 (r + 1)5/2 2 + (α 3 − 6α 2 + 11α − 6)r2 r2 + 1 + (α 3 − 32 α 2 − 4α − 32 ) r2 + 1 ] α hα [ 32 (α 2 + 1)( r2 + 1 + r)3 + 32 (α 2 − 9)( r2 + 1 + r) = 2 5/2 (r + 1) + (α − 3)(α − 2)(α − 1)r2 r2 + 1 + (α − 3)(α 2 − 4) r2 + 1 ], which is positive on R for α ≥ 3. This shows (c) for n = 3. Remark 2.20. Open question: In the notation of Example 2.19, is hα monotone of order α + 1 for every α > 0? Some transformations more general than the fractional integral I+p σ are studied by Maejima, Pérez-Abreu, and Sato [24], which is related to [23].

3 Preliminaries in Probability Theory 3.1 Lévy–Khintchine Representation of Infinitely Divisible Distributions We also use a representation of the cumulant function Cμ (z) of μ ∈ ID other than (1.1) in the form 1 Cμ (z) = − z, Aμ z + 2

iz, x iz,x −1− νμ (dx) + iγμ , z. e 1 + |x|2 Rd

(3.1)

Here γμ is an element of Rd ; Aμ and νμ are common to (1.1) and (3.1). Throughout this article γμ is used in this sense. It follows from (1.1) and (3.1) that

γμ

x|x|2 = γμ − ν (dx) + 2 μ |x|≤1 1 + |x|

|x|>1

x νμ (dx). 1 + |x|2

(3.2)

The triplets (Aμ , νμ , γμ ) and (Aμ , νμ , γμ ) are both called the Lévy–Khintchine triplet of μ . Each has its own advantage and disadvantage. Weak convergence of a sequence of infinitely divisible distributions can be expressed by the corresponding triplets of the type (Aμ , νμ , γμ ), but cannot by the triplets of the type (Aμ , νμ , γμ ). This is because the integrand in the integral term is continuous with respect to x in (3.1), but not continuous in (1.1). On the other hand the formulas derived from


27

(Aμ , νμ , γμ ) are often simpler than those derived from (Aμ , νμ , γμ ). See the book [39] for details. In [39] the author uses the symbol γ in the sense of γμ , but in the papers [40]–[44] in the sense of γμ . The γμ and γμ are both called the location parameter of μ . They depend on the choice of the integrand in the Lévy–Khintchine representation. Many other choices of the integrand are found in the literature. Kwapień and Woyczyński [17] and Rajput and Rosinski [31] use some form other than in (1.1) and (3.1). Maruyama [29] uses still another form.

3.2 Radial and Spherical Decompositions of σ -Finite Measures on Rd A measure ν (B), B ∈ B(Rd ), is called σ -finite if there is a Borel partition Bn , n = 1, 2, . . ., of Rd such that ν (Bn ) < ∞. The following propositions give two decompositions of σ -finite measures on Rd . Proposition 3.1. Let ν be a σ -finite measure on Rd satisfying ν ({0}) = 0. Then there are a σ -finite measure λ on S = {ξ : |ξ | = 1} with λ (S) ≥ 0 and a measurable family {νξ : ξ ∈ S} of σ -finite measures on R◦+ with νξ (R◦+ ) > 0 such that

ν (B) =

S

λ (d ξ )

R◦+

1B (rξ )νξ (dr),

B ∈ B(Rd ).

(3.3)

Here λ and νξ are uniquely determined in the following sense: if (λ (d ξ ), νξ ) and (λ (d ξ ), νξ ) both satisfy these conditions, then there is a measurable function c(ξ ) on S such that 0 < c(ξ ) < ∞, c(ξ )λ (d ξ ) = λ (d ξ ),

(3.4) (3.5)

νξ (dr) = c(ξ )νξ (dr) for λ -a. e. ξ ∈ S.

(3.6)

We call the pair (λ (d ξ ), νξ ) in this proposition a radial decomposition or polar decomposition of ν . Proof of Proposition 3.1. If ν = 0, then λ = 0 and arbitrary νξ satisfy the assertion. Assume that ν = 0. Let Bn , n = 1, 2, . . ., be a Borel partition of Rd \ {0} such let f (x) = 2−n /an for x ∈ Bn . If an = 0, then that an = ν (Bn ) < ∞. If an > 0, then −n let f (x) = 1 for x ∈ Bn . Let b = Rd \{0} f (x)ν (dx). We have 0 < b ≤ ∑∞ n=1 2 . Let ν (dx) = b−1 f (x)ν (dx), which is a probability measure. Using the conditional distribution theorem, we find a probability measure λ on S and a measurable family {ν ξ : ξ ∈ S} of probability measures on R◦+ such that

ν (B) =

S

λ (d ξ )

R◦+

1B (rξ )ν ξ (dr),

B ∈ B(Rd ).

28

K. Sato

Thus

ν (B) =

B

b f (x)−1 ν (dx) =

S

λ (d ξ )

R◦+

1B (rξ )b f (rξ )−1 ν ξ (dr).

λ and νξ (dr) = b f (rξ )−1 ν ξ (dr). Then νξ is a σ -finite measure on R◦+ Let λ = for each ξ and (3.3) holds. To see the uniqueness, let (λ (d ξ ), νξ ) be the pair just constructed, and let (λ (d ξ ), νξ ) be another decomposition of ν . Then, for every E ∈ B(S), λ (E) = λ (E) = ν ((0, ∞)E) = =

E

λ (d ξ )

R◦+

(0,∞)E

b−1 f (x)ν (dx)

b−1 f (rξ )νξ (dr).

Let c(ξ ) = R◦+ b−1 f (rξ )νξ (dr). Then c(ξ ) is positive for all ξ and finite for λ -a. e. ξ . Modify c(ξ ) on a λ -null set so that (3.4) holds. Now we have (3.5). Then (3.6) also follows. It follows that (3.4)–(3.6) hold for two arbitrary decompositions with an appropriate c(ξ ). Remark 3.2. If ν = 0, then we can choose the measure λ in Proposition 3.1 to be a probability measure. Indeed, λ in the proof is a probability measure. Proposition 3.3. Let ν be a σ -finite measure on Rd satisfying ν ({0}) = 0. Then there are a σ -finite measure ν¯ on R◦+ with ν¯ (R◦+ ) ≥ 0 and a measurable family {λr : r ∈ R◦+ } of σ -finite measures on S = {ξ : |ξ | = 1} with λr (S) > 0 such that

ν (B) =

R◦+

ν¯ (dr)

S

1B (rξ )λr (d ξ ),

B ∈ B(Rd ).

(3.7)

Here ν¯ and λr are uniquely determined in the following sense: if (ν¯ (dr), λr ) and (ν¯ (dr), λr ) both satisfy these conditions, then there is a measurable function c(r) on R◦+ such that 0 < c(r) < ∞, c(r)ν (dr) = ν¯ (dr), λr (d ξ ) = c(r)λr (d ξ ) for ν¯ -a. e. r ∈ R◦+ . ¯

(3.8) (3.9) (3.10)

Proof is similar to that of Proposition 3.1, interchanging the roles of S and R◦+ . We call the pair (ν¯ (dr), λr ) in this proposition a spherical decomposition of ν . Remark 3.4. If there is a positive measurable function f (r) on R◦+ such that 3.3 can be chosen to be Rd \{0} f (|x|)ν (dx) < ∞, then λr , r ∈ S, in Proposition probability measures. Indeed, since Rd f (|x|)ν (dx) = R◦+ f (r)λr (S)ν¯ (dr), λr (S) is finite for ν¯ -a. e. r. Noting that

ν (B) =

R◦+

λr (S)ν¯ (dr)

S

1B (rξ )(λr (S))−1 λr (d ξ ),


29

choose ν (dr) = λr (S)ν¯ (dr), λr (d ξ ) = (λr (S))−1 λr (d ξ ) for ν¯ -a. e. r, and λr

¯ appropriately for r in a ν -null set and consider (ν (dr), λr ) as a new spherical decomposition. We say that the Lévy measure νμ of μ ∈ ID is of polar product type if there are a finite measure λμ on S and a σ -finite measure ν¯ μ on R◦+ such that

ν (B) =

S

λμ (d ξ )

R◦+

1B (rξ )ν¯ μ (dr),

B ∈ B(Rd ).

(3.11)

Example 3.5. Any stable distribution μ on Rdhas Lévy measure of polar product type. Indeed, if μ is α -stable, then νμ (B) = S λ (d ξ ) R◦ 1B (rξ )r−α −1 dr with a + finite measure λ on S.

3.3 Weak Mean of Infinitely Divisible Distributions

d As usual, a distribution μ on R is said to haved mean mμ if Rd |x|μ (dx) < ∞ and x μ (dx) = m . We say that μ has mean in R if d μ R Rd |x| μ (dx) < ∞.

Definition 3.6. Let μ ∈ ID. We say that μ has weak mean in Rd if 1 0 such that e−a2 s ≤ ϕ0 (s) ≤ e−a1 s for all large s,

(4.1)

38

K. Sato

and

ϕα (s) s−1/α

with α ∈ (0, ∞) as s → ∞.

(4.2)

In general, for two functions f and g we write f (s) g(s), s → ∞, if there are positive constants a1 and a2 such that 0 < a1 g(s) f (s) a2 g(s) for all large s. The following description of the domains is known. Theorem 4.1. Let 0 ≤ α < ∞. Suppose that ϕα is locally square-integrable on R+ and satisfies (4.1)–(4.2). (i) If α = 0, then D(ΦϕL0 ) = {ν ∈ ML :

|x|>1 log |x|ν (dx)

< ∞}.

(ii) If 0 < α < 2, then D(ΦϕLα ) = {ν ∈ ML :

α |x|>1 |x| ν (dx)

< ∞}.

(iii) If α ≥ 2, then D(ΦϕLα ) = {δ0 }. Proof is given in the same way as that of Lemma 2.7 of [42] and Proposition 4.3 of [43]. A similar proof will be given to Theorem 6.2. Theorem 4.2. Let 0 ≤ α < ∞. Suppose that ϕα is locally square-integrable on R+ and satisfies (4.1)–(4.2). (i) If α = 0, then D0 (Φϕ0 ) = D(Φϕ0 ) = De (Φϕ0 ) = {ρ ∈ ID :

|x|>1 log |x|νρ (dx)

< ∞}. (4.3)

(ii) If 0 < α < 1, then D0 (Φϕα ) = D(Φϕα ) = De (Φϕα ) = {ρ ∈ ID :

α |x|>1 |x| νρ (dx)

< ∞}. (4.4)

(iii) If α = 1, then De (Φϕ1 ) = {ρ ∈ ID : D(Φϕ1 ) = {ρ ∈ ID : t

lim

t→∞ 0

= {ρ ∈ ID : t

lim

t→∞ 0

D (Φϕ1 ) = {ρ ∈ ID : 0

|x|>1 |x|νρ (dx)

< ∞}

|x|>1 |x|νρ (dx)

< ∞,

ds

|ϕ1 (s)x|>1

ds

Rd

Rd x ρ (dx)

= 0,

ϕ1 (s)x νρ (dx) exists in Rd }

|x|>1 |x|νρ (dx)

< ∞,

Rd x ρ (dx)

= 0,

ϕ1 (s)x |ϕ1 (s)x|2 νρ (dx) exists in Rd }, 1 + |ϕ1(s)x|2

|x|>1 |x|νρ (dx)

< ∞,

Rd x ρ (dx)

= 0,


∞ 0

ds |ϕ

1

39

ϕ1 (s)x νρ (dx) < ∞} (s)x|>1

= {ρ ∈ ID : |x|>1 |x|νρ (dx) < ∞, Rd x ρ (dx) = 0, ∞ ϕ1 (s)x |ϕ1 (s)x|2 < ∞}. ds ν (dx) ρ 2 Rd 1 + |ϕ1 (s)x| 0 (iv) If 1 < α < 2, then D0 (Φϕα ) = D(Φϕα ) De (Φϕα ), De (Φϕα ) = {ρ ∈ ID : |x|>1 |x|α νρ (dx) < ∞},

(4.5) (4.6)

D0 (Φϕα ) = D(Φϕα ) = { μ ∈ De (Φϕα ) :

x ρ (dx) = 0}.

(4.7)

D0 (Φϕα ) = D(Φϕα ) = {δ0 } De (Φϕα ) = {δγ : γ ∈ Rd }.

(4.8)

Rd

(v) If α ≥ 2, then

Recall the following. If ρ ∈ ID, then |x|>1 log |x|νρ (dx) < ∞ and |x|>1 log |x| ρ (dx) < ∞ are equivalent. If ρ ∈ ID and α > 0, then |x|>1 |x|α νρ (dx) < ∞ and α Rd |x| ρ (dx) < ∞ are equivalent. See Theorem 25.3 of [39]. Lemma 4.3. Let ρ ∈ ID and Rd

Rd

xρ (dx) = 0

|x|ρ (dx) < ∞. Then ⇔

γρ = −

⇔

γρ

=−

|x|>1

Rd

x νρ (dx)

x|x|2 νρ (dx). 1 + |x|2

(4.9)

Proof. Straightforward from (3.2) and (3.17).

Proof of Theorem 4.2. Except assertion (iii), these are shown in Theorem 2.4 of [42]. A similar proof will be given to Theorem 6.3. Let us prove (iii). For the description of De (Φϕ1 ), combine Proposition 3.18 (iv) with Theorem 4.1. In order to describe D(Φϕ1 ), first note that ρ ∈ D(Φϕ1 ) if and only if ρ ∈ De (Φϕ1 ) and (3.31) holds (Proposition 3.18 (i)). Recall that ϕ1 (s) s−1 as s → ∞. Assume that ρ ∈ D(Φϕ1 ). Then |x|>1 |x|νρ (dx) < ∞. We have 1{|ϕ1 (s)x|≤1} − 1{|x|≤1} → 1{|x|>1} as s → ∞. It follows from the dominated convergence theorem that Rd

x(1{|ϕ1 (s)x|≤1} − 1{|x|≤1} )νρ (dx) →

|x|>1

x νρ (dx),

s → ∞,

since |x(1{|ϕ1 (s)x|≤1} − 1{|x|≤1} )| is bounded by |x| for all x and equals 0 for |x| ≤ 1 and |ϕ1 (s)| ≤ 1. Since γμt of (3.22) is convergent, it follows that γρ satisfies the

40

K. Sato

condition in (4.9). Hence obtain

Rd

xρ (dx) = 0. Replacing γρ in (3.22) by that of (4.9), we t

γμt = −

ds 0

|ϕ1 (s)x|>1

ϕ1 (s)x νρ (dx).

Hence μ belongs to the right-hand side of the first equality of the description of D(Φϕ1 ). The converse direction is similar. This proves the first equality concerning D(Φϕ1 ). The proof of the second equality is done in the same idea. In order to describe D0 (Φϕ1 ), notice that ρ ∈ D0 (Φϕ1 ) if and only if ρ ∈ De (Φϕ1 ) and (3.37) holds (Proposition 3.18 (iii)) and use Lemma 4.3. Finer results are given in the case α = 1. Theorem 4.4. Suppose that ϕ1 is locally square-integrable on R+ and satisfies ϕ1 (s) s−1 as s → ∞. Suppose, in addition, that ∞ 1

|ϕ1 (s) − cs−1 |ds < ∞

(4.10)

with some c > 0. Then D0 (Φϕ1 ) D(Φϕ1 ) De (Φϕ1 ), D(Φϕ1 ) = {ρ ∈ ID : |x|>1 |x|νρ (dx) < ∞, Rd x ρ (dx) = 0, t

lim

t→∞ 1

D0 (Φϕ1 ) = {ρ ∈ ID :

s−1 ds

|x|>s

xνρ (dx) exists in Rd },

|x|>1 |x|νρ (dx)

(4.11) (4.12)

< ∞, Rd x ρ (dx) = 0, ∞ −1 s ds xνρ (dx) < ∞}. |x|>s 1

(4.13)

This is Theorem 2.8 of [42]. For α < 1 description of the ranges is simple. Proposition 4.5. Let 0 ≤ α < 1. Suppose that ϕα is locally square-integrable on R+ and satisfies (4.1)–(4.2). Suppose further that ϕα ≥ 0. Then R0 (Φϕα ) = R(Φϕα ) = Re (Φϕα ) = { μ ∈ ID : νμ ∈ R(ΦϕLα )}.

(4.14)

Proof. It is known in (3.28) that R0 (Φϕα ) ⊂ R(Φϕα ) ⊂ Re (Φϕα ). Suppose that μ ∈ (ρ ) Re (Φϕα ). Then μ is the law of p-lim( 0t ϕα (s)dXs − qt )) for some ρ ∈ De (Φϕα ) t→∞

and some qt . Since α < 1, it follows from Theorem 4.2 that ρ ∈ D0 (Φϕα ). Hence t (ρ ) is convergent in probability as t → ∞. Thus qt tendsto some q ∈ Rd . 0 ϕα (s)dXs

∗ δ−q for some μ

∈ R0 (Φϕα ). Since 0 < 0∞ ϕα (s)ds < ∞, It follows that μ = μ 0 we see that μ itself belongs to R (Φϕα ). The assertion Re (Φϕα ) = {μ ∈ ID : νμ ∈ R(ΦϕLα )} comes from Proposition 3.27.


41

4.2 Φ¯ p,α and Φ¯ p,L α Let −∞ < α < ∞ and 0 < p < ∞. In Section 1.6 we have introduced the twoparameter family of mappings Φ¯ p,α . Namely, starting from the function s = g¯ p,α (t) of (1.13), we define its inverse function t = f¯p,α (s) for 0 ≤ s < a¯ p,α = g¯ p,α (0+), where g¯ p,α (0+) = Γ−α /Γp−α for α < 0 and ∞ for α ≥ 0; if α < 0, then f¯p,α (s) is defined to be zero for s ≥ a¯ p,α ; then we define

Φ¯ p,α ρ = L

∞−

0

(ρ ) f¯p,α (s)dXs

(4.15)

with D(Φ¯ p,α ) being the class of ρ ∈ ID such that the improper stochastic integral in (4.15) is definable. Note the following special cases. If p = 1 and α = 0, then g¯1,0 (t) = − logt,

0 < t ≤ 1,

f¯1,0 (s) = e−s ,

s ≥ 0.

(4.16)

Thus Φ¯ 1,0 = Φ , where Φ is given by (1.11). If p = 1 and α = 0, then g¯1,α (t) =

α −1 (t −α − 1),

0 0,

(−α )−1 (1 − t −α ),

0≤t ≤1

for α < 0,

(1 + α s)−1/α ,

s≥0

f¯1,α (s) =

(1 − (−α )s)1/(−α ),

0≤s

for α > 0, ≤ (−α )−1

for α < 0.

(4.17)

(4.18)

If p > 0 and α = −1, then g¯ p,−1(t) = c p+1 (1 − t) p, 0 ≤ t ≤ 1, f¯p,−1 (s) = 1 − (Γp+1s)1/p , 0 ≤ s ≤ c p+1 .

(4.19) (4.20)

Asymptotic behaviors of f¯p,α (s) for α ≥ 0 are as follows. This is given in Proposition 1.1 of [42] without proof. Proposition 4.6. As s → ∞, f¯p,0 (s) ∼ exp(c − Γp s) for p > 0, f¯p,α (s) ∼ (αΓp s)−1/α for α > 0 and p > 0, f¯p,1 (s) = (Γp s)−1 − (1 − p)(Γp s)−2 log s + O(s−2 ) for p > 0, where c = (p − 1)

1 0

(1 − u) p−2 log u du.

(4.21) (4.22) (4.23)

(4.24)

42

K. Sato

Proof. (4.21): As t ↓ 0, g¯ p,0 (t) = c p

1 t

u−1 du + c p

= −c p logt + c p

1 0

1 t

((1 − u) p−1 − 1)u−1du

((1 − u) p−1 − 1)u−1du + o(1)

and 1 0

((1 − u) p−1 − 1)u−1du = (1 − p) = (1 − p)

1

u−1 du

u

0

1 0

0

(1 − v) p−2dv

1 (1 − v) p−2 log dv = c. v

It follows that s = c p (− log f¯p,0 (s) + c) + o(1),

s → ∞,

that is, (4.21) holds. (4.22): Let α > 0 and p > 0. As t ↓ 0, g¯ p,α (t) = c p =α

1

t −1

u−α −1du + c p

c p t −α + O(t

1

t −α +1

((1 − u) p−1 − 1)u−α −1du

).

Hence s = f¯p,α (s)−α (α −1 c p + o(1)),

s → ∞.

(4.23): Let p > 0. We have g¯ p,1 (t) = c pt −1 − (1 − p)c p logt + O(1),

t ↓ 0,

since

1 −1 p−1 −2 g¯ p,1(t) = c p t + ((1 − u) − 1)u du − 1 t

1 1 −1 p−1 −2 udu + ((1 − u) − 1 − (1 − p)u)u du − 1 . = c p t + (1 − p) t

t

Hence s = c p f¯p,1 (s)−1 − (1 − p)c p log f¯p,1 (s) + O(1),

s → ∞,

that is, f¯p,1 (s) = c p s−1 − (1 − p)c ps−1 f¯p,1 (s) log f¯p,1 (s) + O(s−1 f¯p,1 (s)).

(4.25)


43

On the other hand we have g¯ p,1(t) = c pt −1 + o(t −1), t ↓ 0, s = c p f¯p,1 (s)−1 + o( f¯p,1(s)−1 ),

s → ∞.

f¯p,1 (s) = c p s−1 (1 + o(1)), successively. The last formula and (4.25) yield (4.23) with o(s−2 log s) in place of O(s−2 ). Then this and (4.25) give (4.23). If α < 0, then D0 (Φ¯ p,α ) = D(Φ¯ p,α ) = De (Φ¯ p,α ) = ID(Rd ). If α ≥ 0, then ¯ p,α ), and De (Φ¯ p,α ) are described by Theorems 4.2 and 4.4 by virtue p,α ), D(Φ of Proposition 4.6. As a consequence, they do not depend on p. We notice that Φ¯ p,α is trivial if α ≥ 2. L = Φ L with f = f¯ For −∞ < α < ∞ and p > 0 we define Φ¯ p, p,α as in α f Definition 3.25. Again by Proposition 4.6, Theorem 4.1 is applied to the description L ), which does not depend on p. If α ≥ 2, then Φ L is trivial. If α < 0, ¯ p, of D(Φ¯ p, α α L L then D(Φ¯ p,α ) = M . L ), then If ν ∈ D(Φ¯ p, α D0 (Φ¯

L Φ¯ p, α ν (B) =

∞

ds 0

=− = cp

1 0

1 0

Rd

1B ( f¯p,α (s)x)ν (dx)

d g¯ p,α (t)

Rd

1B (tx)ν (dx)

(1 − t) p−1t −α −1 dt

Rd

(4.26)

1B (tx)ν (dx)

L is an example of ϒ -transformations for B ∈ B(Rd \ {0}). This shows that Φ¯ p, α studied by Barndorff-Nielsen, Rosiński, and Thorbjørnsen [2]. L The family Φ¯ p, α satisfies the following identity.

Theorem 4.7. Let −∞ < α < 2, p > 0, and q > 0. Then L ¯L ¯L ¯L ¯L Φ¯ p+q, α = Φq,α −p Φ p,α = Φ p,α Φq,α −p .

(4.27)

Proof. Let ν ∈ ML (Rd ). Let ν ( j) , j = 1, 2, 3, 4, be measures on Rd with ν ( j) ({0}) = 0 satisfying

ν (1) (B) = ν (2) (B) = ν (3) (B) = ν (4) (B) =

∞

ds 0

∞

Rd

Rd

Rd

ds 0

∞

ds 0

∞

ds 0

Rd

1B ( f¯p,α (s)x)ν (dx), 1B ( f¯q,α −p (s)x)ν (1) (dx), 1B ( f¯q,α −p (s)x)ν (dx), 1B ( f¯p,α (s)x)ν (3) (dx)

44

K. Sato

for B ∈ B(Rd \ {0}). Then

ν (2) (B) = cq

1 0

= cq c p = cq c p = cq c p = cq c p = cq c p

= c p+q

(1 − t)q−1t −α +p−1dt

1

0

(1 − t)q−1t −α +p−1dt

Rd Rd Rd Rd

1 0

ν (dx) ν (dx) ν (dx) ν (dx)

1 0

1 0

1 0

1 0

1B (tx)ν (1) (dx)

Rd

1 0

(1 − u) p−1u−α −1 du

(1 − t)q−1t −α +p−1dt (1 − t)q−1dt

t 0

1 0

Rd

1B (tux)ν (dx)

1B (tux)(1 − u) p−1u−α −1 du

1B (wx)(t − w) p−1w−α −1 dw

1B (wx)w−α −1 dw

1 w

(1 − t)q−1(t − w) p−1 dt

1B (wx)(1 − w) p+q−1w−α −1 dw

1 0

(1 − y)q−1y p−1 dy

(by change of variables y = (t − w)/(1 − w)) (1 − w) p+q−1w−α −1 dw

1B (wx)ν (dx).

Rd

Hence it follows from Definition 3.25 that

ν (2) ∈ ML (Rd )

L ν ∈ D(Φ¯ p+q, α ).

⇔

On the other hand,

ν (2) ∈ ML (Rd )

ν (1) ∈ D(Φ¯ q,L α −p )

⇔

and L ν ∈ D(Φ¯ p+q, α)

⇔

L ν ∈ D(Φ¯ p, α)

by Proposition 4.6. Hence L ν ∈ D(Φ¯ p+q, α)

⇔

L ¯L ¯L ν ∈ D(Φ¯ p, α ), Φ p,α ν ∈ D(Φq,α −p )

L ¯L ¯L ¯L ¯L ¯L and Φ¯ p+q, α = Φq,α −p Φ p,α . Similarly, in order to see Φ p+q,α = Φ p,α Φq,α −p , observe that

ν (4) (B) = c p = c p cq

1

1 0

0

(1 − u) p−1u−α −1 du

(1 − u) p−1u−α −1 du

1 0

Rd

1B (ux)ν (3) (dx)

(1 − t)q−1t −α +p−1dt

which equals ν (2) (B) in the preceding calculus.

Rd

1B (utx)ν (dx),


45

Corollary 4.8. We have L ¯L R(Φ¯ p, α ) ⊃ R(Φ p ,α )

R(Φ¯ αL −β ,α )

for 0 < p < p < ∞ and −∞ < α < 2,

⊃ R(Φ¯ αL −β ,α )

for − ∞ < β < α < α < 2.

(4.28) (4.29)

L Φ ¯L Proof. The decrease (4.28) follows from Φ¯ pL ,α = Φ¯ p, α p −p,α −p in (4.27). If L L L −∞ < β < α < α < 2, then Φ¯ α −β ,α = Φ¯ α −β ,α Φ¯ α −α ,α . Hence the decrease (4.29) follows.

4.3 Range of Φ¯ p,L α L ) and the one-to-one property of Φ L . ¯ p, Let us give the description of R(Φ¯ p, α α

Theorem 4.9. Let −∞ < α < 2 and 0 < p < ∞. L ) with a radial decomposition (λ (d ξ ), ν ) and let ν L ν.

= Φ¯ p, (i) Let ν ∈ D(Φ¯ p, ξ α α − α −1 Then ν has a radial decomposition (λ (d ξ ), u kξ (u)du), where

kξ (u) = c p

(u,∞)

(r − u) p−1rα −p+1 νξ (dr).

(4.30)

L is one-to-one. (ii) Φ¯ p, α

Proof. Let us show (i). Beginning with (4.26) we have, for B ∈ B(Rd \ {0}),

ν (B) = c p = cp =

S

S

S

λ (d ξ ) λ (d ξ )

λ (d ξ )

(0,∞)

νξ (dr)

(0,∞)

(0,∞)

r

1

α −p+1

0

(1 − t) p−1t −α −1 1B (trξ )dt

νξ (dr)

r 0

(r − u) p−1 u−α −1 1B (uξ )du

1B (uξ )u−α −1kξ (u)du,

where kξ (u) is given by (4.30). Since νξ (R◦+ ) > 0 for each ξ by the definition of a radial decomposition, kξ (u) is not identically zero for each ξ . L ). Let ν L ν = Φ L ν for some ν , ν ∈ ¯ p,

= Φ¯ p, To prove (ii), let ν ∈ R(Φ¯ p, α α α L ). Let (λ (d ξ ), ν ) and (λ (d ξ ), ν ) be radial decompositions of ν and D(Φ¯ p, ξ α ξ

ν , respectively. Then ν has radial decompositions (λ (d ξ ), u−α −1kξ (u)du) and (λ (d ξ ), u−α −1 kξ (u)du), where kξ (u) is given by (4.30) and kξ (u) is given by (4.30) with νξ in place of νξ . Hence, by Proposition 3.1, there is a measurable function c(ξ ) satisfying (3.4), (3.5), and u−α −1 kξ (u)du = c(ξ )u−α −1kξ (u)du for λ -a. e. ξ . Hence kξ (u)du = c(ξ )kξ (u)du, λ -a. e. ξ .

46

K. Sato

Since ν ∈ ML (Rd ), kξ (u)du and kξ (u)du are, for λ -a. e. ξ , locally finite measures on R◦+ . Therefore, Theorem 2.10 on the one-to-one property of I+p guarantees that rα −p+1νξ (dr) = c(ξ )rα −p+1νξ (dr),

λ -a. e. ξ .

Hence νξ = c(ξ )νξ (dr), λ -a. e. ξ , and we obtain ν = ν .

L is characterized, using the notion of monotonicity of order p. The range of Φ¯ p, α

Theorem 4.10. Let −∞ < α < 2 and 0 < p < ∞. A measure η on Rd belongs L ) if and only if η is in ML and has a radial decomposition (λ (d ξ ), to R(Φ¯ p, α u−α −1 kξ (u)du) such that kξ (u) is measurable in (ξ , u) and, for λ -a. e. ξ , monotone of order p on R◦+ in u.

(4.31)

L ). Then η ∈ ML and η = Φ L ν for some ν ∈ D(Φ L ). ¯ p, ¯ p, Proof. Let η ∈ R(Φ¯ p, α α α Thus by Theorem 4.9 η has a radial decomposition (λ (d ξ ), u−α −1 kξ (u)du) with kξ (u) of (4.30). Since {νξ } is a measurable family, kξ (u) is measurable in (ξ , u). Since kξ (u)du and rα −p+1 νξ (dr) are locally finite measures on R◦+ for λ -a. e. ξ , (4.30) shows that rα −p+1 νξ (dr) ∈ D(I+p ) for λ -a. e. ξ and that kξ (u) is monotone of order p on R◦+ for λ -a. e. ξ . Conversely, suppose that η ∈ ML with a radial decomposition (λ (d ξ ), −α −1 u kξ (u)du) satisfying (4.31). Modifying kξ (u) for ξ in a λ -null set, we can assume that, for all ξ , kξ (u) is monotone of order p on R◦+ . From the definition of monotonicity of order p, there is a measure σξ ∈ M∞p−1 (R◦+ ) such that

kξ (u) = c p

(u,∞)

(r − u) p−1σξ (dr).

It follows from Proposition 2.16 that {σξ : ξ ∈ S} is a measurable family. Let νξ (dr) = r−α +p−1σξ (dr). Define ν by ν (B) = S λ (d ξ ) (0,∞) 1B (rξ )νξ (dr). Then the equalities in the proof of Theorem 4.9 (i) show that

η (B) =

∞

ds 0

Rd

1B ( f¯p,α (s)x)ν (dx)

L for B ∈ B(Rd \ {0}). Since η ∈ ML , it follows from Remark 3.26 that ν ∈ D(Φ¯ p, α) L ν. and η = Φ¯ p, α


47

4.4 Classes Kp,α , Kp,0 α , and Kp,e α For −∞ < α < 2 and p > 0 we define K p,α = K p,α (Rd ) = R(Φ¯ p,α ), 0 0 d 0 ¯ K p, α = K p,α (R ) = R (Φ p,α ),

(4.33)

e e d e ¯ K p, α = K p,α (R ) = R (Φ p,α ).

(4.34)

(4.32)

Proposition 4.11. We have 0 e K p, α = K p,α = K p,α 0 K p,1 0 K p, α

e ⊂ K p,1 ⊂ K p,1 , e = K p,α ⊂ K p,α

for −∞ < α < 1,

(4.35) (4.36)

for 1 < α < 2.

(4.37)

Proof. Use Proposition 4.6. If α < 0, then (4.35) comes from a¯ p,α < ∞. If 0 ≤ α < 1, then (4.35) comes from Proposition 4.5. We have (4.36) from (3.28). If 1 < α < 2, then we have (4.37) from (3.28) and Theorem 4.2. In Section 5 of [42] it is conjectured that, in the notation of the present article,

Φ¯ p+q,α = Φ¯ q,α −p Φ¯ p,α = Φ¯ p,α Φ¯ q,α −p

(4.38)

for α ∈ R, p > 0, and q > 0. For ρ ∈ D0 (Φ¯ p+q,α ) these equalities are proved. Theorem 4.12. Let −∞ < α < 2, p > 0, and q > 0. Let ρ ∈ D0 (Φ¯ p+q,α ) = D0 (Φ¯ p,α ). Then

ρ ∈ D0 (Φ¯ q,α −p ), Φ¯ q,α −p ρ ∈ D0 (Φ¯ p,α ), Φ¯ p,α ρ ∈ D0 (Φ¯ q,α −p ),

(4.39)

and

Φ¯ p+q,α ρ = Φ¯ q,α −p Φ¯ p,α ρ = Φ¯ p,α Φ¯ q,α −p ρ .

(4.40)

Proof. A distribution ρ ∈ ID is in D0 (Φ¯ p+q,α ) if and only if 1

c p+q

0

|Cρ (tz)|(1 − t) p+q−1t −α −1 dt < ∞.

(4.41)

Thus, (4.39) holds if and only if 1

cq

cp cq

0 1 0

|Cρ (tz)|(1 − t)q−1t −α +p−1dt < ∞,

(4.42)

|CΦ¯ q,α −p ρ (tz)|(1 − t) p−1t −α −1dt < ∞,

(4.43)

|CΦ¯ p,α ρ (tz)|(1 − t)q−1t −α +p−1dt < ∞.

(4.44)

0 1

48

K. Sato

We assume ρ ∈ D0 (Φ¯ p+q,α ) = D0 (Φ¯ p,α ), that is, (4.41). Then (4.42) holds, 1 1/2 since 1/2 |Cρ (tz)|(1 − t)q−1 dt < ∞ as |Cρ (tz)| is bounded and since 0 |Cρ (tz)| t −α +p−1dt < ∞ from (4.41). To see (4.43), notice that the quantity in (4.43) is = c p cq ≤ c p cq = c p+q

1 0

1 0

(1 − u)

u

1 q−1 −α +p−1 du Cρ (tuz)(1 − t) t dt 0

(1 − u) p−1u−α −1 du

1 0

p−1 −α −1

1 0

|Cρ (tuz)|(1 − t)q−1t −α +p−1dt

|Cρ (vz)|(1 − v) p+q−1v−α −1 dv,

where the last equality is obtained in the proof of Theorem 4.7. (4.44) is similarly true, since the quantity in (4.44) is ≤ cq c p

1 0

(1 − t)q−1t −α +p−1dt

1 0

|Cρ (utz)|(1 − u) p−1u−α −1 du.

Hence (4.39) is true. Now, the estimate above guarantees the use of Fubini’s theorem in showing that 1

c p cq

0

(1 − u) p−1u−α −1du

= c p+q

1 0

1 0

Cρ (tuz)(1 − t)q−1t −α +p−1dt

Cρ (vz)(1 − v) p+q−1v−α −1 dv.

Thus we obtain (4.40).

Remark 4.13. By the method of the proof of Theorem 7.3 (ii) in Section 7.1, we can prove that (4.38) holds if −∞ < α < 1, p > 0, and q > 0, or if 1 < α < 2, 0 0. 0 , and K e . Now we present some decrease properties for K p,α , K p, α p,α

Corollary 4.14. (i) For 0 < p < p < ∞ and −∞ < α < 2, 0 0 K p, α ⊃ K p ,α

e e and K p, α ⊃ K p ,α .

(4.45)

(ii) For −∞ < β < α < α < 2, Kα −β ,α ⊃ Kα −β ,α ,

Kα0 −β ,α ⊃ Kα0 −β ,α ,

and Kαe −β ,α ⊃ Kαe −β ,α . (4.46)

0 , use Theorem 4.12 and proceed as in the proof of Corollary Proof. Concerning K p, α e 4.8. Concerning K p,α , use Proposition 3.27 and Corollary 4.8. Concerning Kα −β ,α ⊃ Kα −β ,α in (ii), it is a consequence of Kα0 −β ,α ⊃ Kα0 −β ,α if α = 1 and α = 1, since 0 0 we have Proposition 4.11. If α = 1, then α > 1 and K1−β ,1 ⊃ K1− β ,1 ⊃ Kα −β ,α = e Kα −β ,α . If α = 1, then α < 1 and Kα −β ,α = Kαe −β ,α ⊃ K1− β ,1 ⊃ K1−β ,1 .


49

The decrease property in (i) is true also for K p,α , but we have to use the later Theorem 4.18. e is as follows. The characterization of K p, α e if and only if μ ∈ ID and Theorem 4.15. Let −∞ < α < 2 and p > 0. Then μ ∈ K p, α its Lévy measure νμ has a radial decomposition (λ (d ξ ), u−α −1kξ (u)du) satisfying

kξ (u) is measurable in (ξ , u) and, for λ -a. e. ξ , monotone of order p on R◦+ in u. Proof. This follows from Proposition 3.27 and Theorem 4.10 immediately.

(4.47)

e . Then Proposition 4.16. Let 0 < α < 2, p > 0, and μ ∈ K p, α

|x|>1

|x|β μ (dx) < ∞ for all β ∈ (0, α ).

(4.48)

L ). So ν = Φ L ν for some ν ∈ D(Φ L ) ¯ p, ¯ p, Proof. The Lévy measure νμ is in R(Φ¯ p, μ α α α and

|x|>1

|x|β νμ (dx) = c p = cp

1 0

(1 − t) p−1t −α −1 dt

|x|>1

≤ const

|x|β ν (dx)

|x|>1

1 1/|x|

|tx|>1

|tx|β ν (dx)

(1 − t) p−1t −1+β −α dt

|x|α ν (dx) < ∞

from Theorem 4.1. Hence we have (4.48).

Remark 4.17. Let 0 < α < 2 and p > 0.

0 such that α (i) There is μ ∈ K p, α Rd |x| μ (dx) = ∞. 0 (ii) There is μ ∈ K p,α which is not Gaussian and satisfies Rd |x|α μ (dx) < ∞ for all α > 0.

These facts follow from Proposition 5.13 combined with Theorem 5.11 of the later section. Characterization of K p,α is as follows. Theorem 4.18. Let −∞ < α < 2 and p > 0. Let μ ∈ ID. (i) Assume that α < 1. Then μ ∈ K p,α if and only if νμ has a radial decomposition (λ (d ξ ), u−α −1 kξ (u)du) satisfying (4.47). (ii) Assume that α = 1. Then μ ∈ K p,1 if and only if μ has the following two properties: νμ has a radial decomposition (λ (d ξ ), u−2 kξ (u)du) satisfying (4.47) with α = 1 and μ has weak mean 0.

50

K. Sato

(iii) Assume that 1 < α < 2. Then μ ∈ K p,α if and only if μ has the following two properties: νμ has a radial decomposition (λ (d ξ ), u−α −1kξ (u)du) satisfying (4.47) and μ has mean 0. Proof. To show (i) for α < 1, recall (4.35). Then the assertion follows from Proposition 3.27 combined with Theorem 4.10. Proof of (ii) for α = 1. Let f = f¯p,1 . The “only if” part. Assume that μ ∈ K p,1 . e from (4.36). Hence ν There is ρ ∈ D(Φ¯ p,1 ) such that μ = Φ¯ p,1 ρ . We have μ ∈ K p,1 μ −2 has a radial decomposition (λ (d ξ ), u kξ (u)du) with kξ (u) satisfying (4.47) with α = 1 by Theorem 4.15. We have |x|>1 |x|νρ (dx) < ∞ and Rd xρ (dx) = 0 from Theorem 4.2 (iii). Hence γρ = − |x|>1 x νρ (dx) from Lemma 4.3. This, combined with (3.22) and (3.35), gives −γμ = lim

t

t→∞ 0

ds

| f (s)x|>1

f (s)x νρ (dx).

(4.49)

Hence −γμ = lim Jε , where Jε = c p

1

ε ↓0

ε

(1 − t) p−1t −1 dt

|x|>1/t

x νρ (dx).

(4.50)

The statement that μ has weak mean 0 is equivalent to the statement that lim Iε exists and equals −γμ , where Iε =

ε ↓0

11 |x|νρ (dx) < ∞, we can use the dominated convergence theorem to conclude (4.52). L ) and μ has weak mean 0. There The “if” part. We have νμ ∈ R(Φ¯ p,1 L ) such that ν = Φ ¯ L ν . We have is ν ∈ D(Φ¯ p,1 μ |x|>1 |x|ν (dx) < ∞ from p,1 ∞ 2 f (s) ds < ∞. Define A by A = ( 0∞ f (s)2 ds)−1 Aμ and Theorem 4.1. We have 0 < 0 γ = − |x|>1 xν (dx). Choose ρ ∈ ID having triplet (Aρ , νρ , γρ ) = (A, ν , γ ). We claim that ρ ∈ D(Φ¯ p,1 ) and Φ¯ p,1 ρ = μ . Since (3.33) and (3.34) hold, it is enough to show (3.35), that is, γμt of (3.22) converges to γμ . Hence it is enough to show (4.50). But we have (4.51), since μ has weak mean 0. The argument in the proof of the “only if ” part proves (4.52). We obtain (4.50) from (4.51) and (4.52) combined. Proof of (iii) for 1 < α < 2. The “only if” part. Let f = f¯p,α . Similarly to the (λ (d ξ ), u−α −1 kξ (u)du) satproof of (i) and (ii), νμ has a radial decomposition α isfying (4.47). We have |x|>1 |x| νρ (dx) < ∞ and Rd xρ (dx) = 0 from Theorem 4.2 since ρ ∈ D0 (Φ¯ p,α ). Thus γρ = − |x|>1 x νρ (dx). Hence we have (4.49), from (3.22) and (3.35). It follows from Proposition 4.16 that |x|>1 |x| νμ (dx) < ∞, that is, ∞ 0 ds | f (s)x|>1 | f (s)x|νρ (dx) < ∞. Therefore

γμ = −

∞

ds 0

| f (s)x|>1

f (s)x νρ (dx) = −

|x|>1

x νμ (dx).

Hence μ has mean 0. The “if” part. Using the argument above, it is not hard to modify the proof of the “if ” part of (ii).

52

K. Sato

Corollary 4.19. For 0 < p < p < ∞ and −∞ < α < 2, K p,α ⊃ K p ,α .

(4.53)

Proof. This follows from Theorem 4.18 and Corollary 2.6. The relation of K p,α and α = 1.

e K p, α

is different in the case 1 < α < 2 and in the case

e can be shifted to an element Corollary 4.20. (i) If 1 < α < 2, then any μ ∈ K p, α of K p,α . e such that, for any x ∈ Rd , the shift μ ∗ δ of μ (ii) If α = 1, then there is μ ∈ K p,1 x does not belong to K p,1 .

Proof. The assertion (i) is clear from Theorems 4.15 and 4.18. To see (ii), let λ be a finite measure on S such that S ξ λ (d ξ ) = 0 and let

ν (B) =

S

λ (d ξ )

∞ 2

1B (rξ )

dr , r2 (log r)1+q

B ∈ B(Rd \ {0})

L L with 0 < q ≤ 1. Then |x|>2 |x|ν (dx) < ∞ and hence ν ∈ D(Φ¯ p,1 ). Let ν = Φ¯ p,1 ν. e Let μ ∈ ID be such that νμ = ν and Aμ and γμ are arbitrary. Then μ ∈ K p,1 . But μ ∈ K p,1 , since

|x|>1/t

xν (dx) =

S

ξ λ (d ξ )

∞ 1/t

dr = q−1 r(log r)1+q

S

ξ λ (d ξ ) (log(1/t))−q

for t < 1/2 and Jε in (4.50) is not convergent as ε ↓ 0. In order to characterize = K p,α for α = 1.

0 K p, α,

we have only to deal with the case α = 1, since

0 K p, α

0 if and only if μ has the followTheorem 4.21. Let p > 0. Let μ ∈ ID. Then μ ∈ K p,1 ing two properties: νμ has a radial decomposition (λ (d ξ ), u−2 kξ (u)du) satisfying (4.47) with α = 1 and μ has weak mean 0 absolutely. 0 , that is, μ = Φ ¯ p,1 ρ for some ρ ∈ Proof. The “only if” part. Assume μ ∈ K p,1 0 L ¯ ¯ νμ ∈ R(Φ p,1 ) and μ has weak mean 0 from D (Φ p,1 ). Then μ ∈ K p,1 . We have Theorem 4.18. We also have |x|>1 |x|νρ (dx) < ∞ and Rd xρ (dx) = 0. Hence γρ = − |x|>1 xνρ (dx). Hence condition (3.37) is written as

∞ 0

ds f (s)

| f (s)x|>1

with f = f¯p,1 , which is equivalent to 1

cp

0

(1 − t)

p−1 −1

t

xνρ (dx) < ∞

dt xνρ (dx) < ∞. |x|>1/t

(4.54)


Let J = cp

1

t

−1

0

53

dt xνρ (dx) . |x|>1/t

Condition (4.54) is equivalent to J < ∞, since 1 1/2

(1 − t) p−1dt

xνρ (dx) ≤

|x|>1/t

1

1/2

(1 − t) p−1dt

|x|>1

|x|νρ (dx) < ∞.

ρ

ρ

Let (ν¯ ρ (dr), λr ) be a spherical decomposition of νρ such that λr , r ∈ R◦+ , are probability measures on S. For each B ∈ B(Rd \ {0}),

νμ (B) =

∞

ds 0

= cp = cp = cp = cp

1

0

Rd

1B ( f (s)x)νρ (dx) = c p

(1 − t) p−1t −2 dt

R◦+ R◦+

∞

ν¯ ρ (dr)

1 0

(u,∞)

0

R◦+

ν¯ ρ (dr)

(1 − t) p−1t −2 dt

r2−p ν¯ ρ (dr) u−2 du

r

1 0

S

S

(1 − t)

p−1 −2

t

dt

Rd

1B (tx)νρ (dx)

1B (trξ )λrρ (d ξ ) 1B (trξ )λrρ (d ξ )

(r − u) p−1u−2 du 1B (uξ )λrρ (d ξ ) S 0

(r − u) p−1r2−p 1B (uξ )λrρ (d ξ ) ν¯ ρ (dr). S

Assuming that νρ = 0, define

λuμ (E) = c p

(u,∞)

(r − u) p−1r2−p λrρ (E)ν¯ ρ (dr),

E ∈ B(S).

μ

μ

Then {λu : u ∈ R◦+ }is a measurable family of measures on S such that λu (S) < ∞ μ for a. e. u > 0, since ε∞ u−2 λu (S)du = |x|>ε νμ (dx) < ∞ for ε > 0. We have now

νμ (B) =

∞ 0

u−2 du

S

and

λuμ (S) = c p

1B (uξ )λuμ (d ξ ), (u,∞)

B ∈ B(Rd \ {0}),

(r − u) p−1 r2−p ν¯ ρ (dr).

We have ν¯ ρ (R◦+ ) > 0 from νρ = 0. Let a = sup{r ∈ R◦+ : ν¯ ρ ((r, ∞)) > 0}. If a = ∞, μ μ μ then λu (S) > 0 for all u ∈ R◦+ . If a 0 for u < a and λu (S) = 0 μ for u ≥ a, and hence νμ ({|x| ≥ a}) = [a,∞) u−2 λu (S)du = 0. Let

ν¯ μ (du) = u−2 1(0,a) (u)du.

54

K. Sato μ

μ

Then, redefining λu appropriately for u in a ν¯ μ -null set, we see that (ν¯ μ (du), λu ) is a spherical decomposition of νμ . Let I=

a∨1 1

u−1 du ξ λuμ (d ξ ) . S

We have u−1 du (r − u) p−1 r2−p ν¯ ρ (dr) ξ λrρ (d ξ ) 1 (u,∞) S 1 = cp t −1 dt (r − 1/t) p−1r2−p ν¯ ρ (dr) ξ λrρ (d ξ ) 0 (1/t,∞) S 1 = cp t −1 dt (1 − 1/(rt)) p−1rν¯ ρ (dr) ξ λrρ (d ξ ) .

I = cp

∞

(1/t,∞)

0

We claim that

S

I 0. Then K p,α

p ∈(p,∞)

K p ,α ,

0 K p, α

p ∈(p,∞)

e K p0 ,α , and K p, α

p ∈(p,∞)

K pe ,α . (4.57)

(ii) If −∞ < β < α < 2, then Kα −β ,α

α ∈(α ,2)

and Kαe −β ,α

Kα −β ,α ,

α ∈(α ,2)

Kα0 −β ,α

Kαe −β ,α .

α ∈(α ,2)

Kα0 −β ,α , (4.58)

Proof. It remains only to show the inclusions are strict. Let us prove (i). Let a ∈ R◦+ and k(u) = (a − u) p−11(0,a)(u). Then k(u) is monotone of order p on R◦+ , but not of order p on R◦+ for any p > p (Example 2.17 (a)). We have 0∞ (u2 ∧ 1)u−α −1k(u)du < ∞. Let λ be a nonzero finite measure on S. Then the measure ν of polar product type (λ (d ξ ), u−α −1 k(u)du) is in L ) \ R(Φ ¯ L ) for any p > p. This shows the third relation in (4.57) and the R(Φ¯ p, α p ,α first and second for α < 1. Noting that |x|>1 |x|ν (dx) < ∞, consider μ ∈ ID with νμ = ν , Aμ arbitrary, and γμ = − |x|>1 xν (dx). Then μ has mean 0 and we obtain the first and second in (4.57) from Theorems 4.18 and 4.21. Proof of (ii). Let −∞ < β < α < α < 2. Let us construct a measure ν in R(Φ¯ αL −β ,α ) \ R(Φ¯ αL −β ,α ) independent of α . For this, let ν be the measure with radial decomposition (λ (d ξ ), u−α −1 e−u du) where λ is a nonzero finite measure. Then ν ∈ R(Φ¯ αL −β ,α ), since e−u is completely monotone. Define l(u) by

u−α −1 e−u = u−α −1 l(u). Then l(u) = uα −α e−u → 0 as u ↓ 0. Hence l(u) is not monotone of finite order on R◦+ , as seen from Proposition 2.13 (iv). Hence ν ∈ R(Φ¯ αL −β ,α ). The rest of the proof is the same as that of (i) We add the one-to-one property of Φ¯ p,α . Theorem 4.23. Let −∞ < α < 2 and p > 0. The mapping Φ¯ p,α is one-to-one. Proof. Suppose that ρ , ρ ∈ D(Φ¯ p,α ) satisfy Φ¯ p,α ρ = Φ¯ p,α ρ . Then, by (3.34) of L ¯L Proposition 3.18, Φ¯ p, 4.9 (ii). α νρ = Φ p,α νρ . Hence νρ = νρ follows from Theorem We have also Aρ = Aρ from (3.33) of Proposition 3.18, since 0 < 0∞ f (s)2 ds < ∞, where we write f = f¯p,α . It follows from (3.22), (3.35), and νρ = νρ that

56

K. Sato

t

x(1{| f (s)x|≤1} − 1{|x|≤1})νρ (dx)

f (s)ds γρ + Rd t = lim f (s)ds γρ +

lim

t→∞ 0

t→∞ 0

Rd

Hence

t

lim

t→∞ 0

x(1{| f (s)x|≤1} − 1{|x|≤1})νρ (dx) .

f (s)(γρ − γρ )ds = 0.

Recall that f (s) > 0 for 0 < s < a¯ p,α . Now we obtain γρ − γρ = 0 irrespective of whether 0∞ f (s)ds is finite or infinite. Therefore ρ = ρ .

e The continuity property of distributions in K p, α is as follows. e with p > 0 and Theorem 4.24. (i) Let μ be a nondegenerate distribution in K p, α α ≥ 0. Then μ is absolutely continuous with respect to d-dimensional Lebesgue measure. (ii) Let μ = Φ¯ p,α ρ with p > 0, α < 0, and ρ ∈ D(Φ¯ p,α ). Then νμ is a finite measure if and only if νρ is a finite measure. In particular, for any p > 0 and α < 0, K p,α contains some compound Poisson distribution, which necessarily has a point mass at the origin.

Here “μ is nondegenerate” means that the support of μ is not a subset of any translation of any (d − 1)-dimensional linear subspace of Rd . This theorem generalizes the fact in [38] that nondegenerate selfdecomposable distributions on Rd are absolutely continuous. L ν0 Proof of Theorem 4.24. Let us show (i). The Lévy measure νμ satisfies νμ = Φ¯ p, α L ). Let (λ (d ξ ), ν 0 (dr)) be a radial decomposition of ν 0 . Then for some ν 0 ∈ D(Φ¯ p, α ξ νμ has a radial decomposition (λ (d ξ ), u−α −1 kξ (u)du) satisfying (4.30) (Theorem 4.9). We have

∞ 0

u−α −1 kξ (u)du = c p = cp

∞

u−α −1du

0

(0,∞)

(u,∞)

(r − u) p−1rα −p+1 νξ0 (dr)

rα −p+1 νξ0 (dr)

r 0

u−α −1 (r − u) p−1du = ∞,

since α ≥ 0 and νξ0 (R◦+ ) > 0. That is, νμ is radially absolutely continuous and satisfies the divergence condition in the sense of [39]. Hence μ is absolutely continuous on Rd by Theorem 27.10 of [39]. L Proof of (ii). We have νμ = Φ¯ p, α νρ . Then it follows from (4.26) that

νμ (Rd ) = c p

1 0

(1 − t) p−1t −α −1 dt νρ (Rd ) = (Γ−α /Γp−α )νρ (Rd ).

Hence the assertion is obvious.


57

5 One-Parameter Subfamilies of {Kp,α } 5.1 Kp,α , K0p,α , and Kep,α for p ∈ (0, ∞) with Fixed α As is shown in Theorem 4.22, the one-parameter families {K p,α : p ∈ (0, ∞)}, 0 : p ∈ (0, ∞)}, and {K e : p ∈ (0, ∞)} for fixed α ∈ (−∞, 2) are strictly de{K p, p,α α creasing as p increases. The limiting classes as p → ∞ are denoted by

K∞,α =

K p,α ,

(5.1)

0 K p, α,

(5.2)

e K p, α.

(5.3)

p>0

0 K∞, α =

p>0

e K∞, α =

p>0

In order to analyze these classes, we use the mappings Ψα , α ∈ R, defined in Section 1.6 from gα (t) and fα (s). For α ≥ 0, fα (s) is positive for all s > 0. For α < 0 we have fα (s) = 0 for s ≥ Γ−α . Asymptotic behaviors of fα (s) are as follows. Proposition 5.1. As s ↓ 0, fα (s) ∼ − log s for α ∈ R.

(5.4)

As s → ∞, f0 (s) ∼ exp(c − s), −1/α

fα (s) ∼ (α s) −1

f1 (s) = s where c=

∞ 1

−2

−s

u−1 e−u du −

(5.5) for α > 0, −2

log s + O(s ),

1 0

u−1 (1 − e−u)du.

(5.6) (5.7)

(5.8)

Proof. Since gα (t) ∼ t −α −1e−t , t → ∞, we have lim s↓0

fα (s) t 1 = lim = lim −α −1 −t = 1, t→∞ t→∞ log(1/s) log(1/gα (t)) t e /gα (t)

that is, (5.4) holds. To see (5.5), note that g0 (t) =

1 t

−1

u du +

1 t

−1

u (e

−u

− 1)du +

∞ 1

u−1 e−u du = − logt + c + o(1)

58

K. Sato

as t ↓ 0 and hence s = − log f0 (s) + c + o(1), s → ∞. To see (5.6), see that gα (t) = α −1t −α (1 + o(1)), t ↓ 0, equivalently, s = α −1 fα (s)−α (1 + o(1)), s → ∞. Assertion (5.7): We have g1 (t) =

∞

=t

t −1

u−2 du +

1 t

u−2 (e−u − 1 + u)du −

+ logt + O(1),

1 t

u−1 du +

∞ 1

u−2 (e−u − 1)du

t↓0

and hence s = f1 (s)−1 + log f1 (s) + O(1), s → ∞, which is written to f1 (s) = s−1 + s−1 f1 (s) log f1 (s) + O(s−1 f1 (s)),

s → ∞.

(5.9)

Since f1 (s) = s−1 (1 + o(1)) from (5.6), we obtain from (5.9) f1 (s) = s−1 − s−2 log s + o(s−2 log s),

s → ∞.

Putting this again in (5.9), we arrive at (5.7).

We define fα (0) = ∞ for convenience. Then fα (s) is locally square-integrable on R+ . We have

Ψα ρ = L

∞− 0

(ρ )

fα (s)dXs

,

that is, Ψα ρ = Φ f ρ with f = fα in (3.24) whenever the improper stochastic integral is definable. If α < 0, then D0 (Ψα ) = D(Ψα ) = De (Ψα ) = ID. By virtue of Proposition 5.1, the domains D0 (Ψα ), D(Ψα ), and De (Ψα ) are given by Theorems 4.2 and 4.4. Thus D0 (Ψα ) = D0 (Φ¯ p,α ), D(Ψα ) = D(Φ¯ p,α ), and De (Ψα ) = De (Φ¯ p,α ) for all α and p. For α ≥ 2, Ψα is trivial. For α = −1 we have g−1 (t) = e−t ,

t ≥ 0,

f−1 (s) = − log s,

0 < s ≤ 1.

(5.10)

Hence Ψ−1 = ϒ , where ϒ is mentioned in Section 1.7. Define ΨαL as Φ Lf in Definition 3.25 with f = fα . This means that

ΨαL ν (B)

=

∞

t 0

−α −1 −t

e dt

Rd

1B (tx)ν (dx)

(5.11)

for B ∈ B(Rd \ {0}). Thus ΨαL is an ϒ -transformation of [2]. We have D(ΨαL ) = L ), which is described by Theorem 4.1. D(Φ¯ p, α The following is an important identity given in Theorem 3.1 of Sato [42] with a long proof. This relates Ψα with Φ¯ p,α . Theorem 5.2. If −∞ < α < 2 and 0 < p < ∞, then

Ψα = Ψα −pΦ¯ p,α = Φ¯ p,α Ψα −p .

(5.12)


59

The prototype of this identity is

Ψ0 = ϒ Φ = Φϒ given in Barndorff-Nielsen, Maejima, and Sato [1]. We will use the following two related facts. Theorem 5.3. Let −∞ < α < 2 and 0 < p < ∞. Suppose that ρ ∈ D0 (Ψα ). Then ρ ∈ D0 (Ψα −p ) ∩ D0 (Φ¯ p,α ), Ψα −p ρ ∈ D0 (Φ¯ p,α ), Φ¯ p,α ρ ∈ D0 (Ψα −p ), and

Ψα ρ = Ψα −pΦ¯ p,α ρ = Φ¯ p,αΨα −p ρ .

(5.13)

This is given in Lemma 3.2 of [42]. Theorem 5.4. If −∞ < α < 2 and 0 < p < ∞, then L L ¯L ΨαL = ΨαL−pΦ¯ p, α = Φ p,α Ψα −p .

(5.14)

Proof. Let ν ∈ ML . Let ν ( j) , j = 1, 2, be measures on Rd with ν ( j) ({0}) = 0 satisfying

ν (1) (B) = ν (2) (B) =

∞

ds 0

∞

ds

Rd Rd

0

1B ( f¯p,α (s)x)ν (dx), 1B ( fα −p (s)x)ν (1) (dx)

for B ∈ B(Rd \ {0}). Then

ν (2) (B) =

∞ 0

= cp = cp = cp = cp =

∞

Rd

Rd

0

Rd

ν (dx) ν (dx) ν (dx)

∞

1B (tx)ν (1) (dx)

Rd

t −α +p−1e−t dt

0

∞


1 0


0

∞

e−t dt

0

∞ 0

v−α −1 e−v dv

(1 − u) p−1u−α −1 du

t 0

1B (vx)v Rd

1 0

On the other hand,

ν (2) ∈ ML

1B (tux)ν (dx)

1B (vx)(t − v) p−1v−α −1 dv

−α −1

∞

dv v

(t − v) p−1e−t dt

1B (vx)ν (dx).

⇔ ⇔

Rd

1B (tux)(1 − u) p−1u−α −1du

Hence

ν (2) ∈ ML

ν ∈ D(ΨαL ). ν (1) ∈ D(ΨαL−p)

60

K. Sato

and

ν ∈ D(ΨαL )

L ν ∈ D(Φ¯ p, α)

⇔

by Propositions 4.6 and 5.1. Hence

ν ∈ D(ΨαL )

⇔

L L ¯L ν ∈ D(Φ¯ p, α ), Φ p,α ν ∈ D(Ψα −p )

L . Proof of Ψ L = Φ L ΨL ¯ p, and ΨαL = ΨαL−pΦ¯ p, α α α α −p is similar.

Theorem 5.5. Let −∞ < α < 2. (i) Let ν ∈ D(ΨαL ) with a radial decomposition (λ (d ξ ), νξ ) and let ν = ΨαL ν . Then ν has a radial decomposition (λ (d ξ ), u−α −1 kξ (u)du), where kξ (u) =

R◦+

rα e−u/r νξ (dr).

(5.15)

(ii) ΨαL is one-to-one. Proof. To see (i), note that it follows from (5.11) that

ν (B) = = =

S

S

S

λ (d ξ ) λ (d ξ ) λ (d ξ )

R◦+

R◦+ ∞ 0

νξ (dr)

∞ 0

rα νξ (dr)

t −α −1 e−t 1B (trξ )dt

∞ 0

u−α −1 e−u/r 1B (uξ )du

1B (uξ )u−α −1du

R◦+

rα e−u/r νξ (dr).

Assertion (ii) is proved from the uniqueness in Bernstein’s theorem on Laplace transforms. See Proposition 4.1 of [42]. Proposition 5.6. Let −∞ < α < 2. The mapping Ψα is one-to-one. This is proved similarly to Theorem 4.23, using the one-to-one property of ΨαL in Theorem 5.5 (ii). Theorem 5.7. Let −∞ < α < 2. A measure η on Rd belongs to R(ΨαL ) if and only if η is in ML and has a radial decomposition (λ (d ξ ), u−α −1kξ (u)du) such that kξ (u) is measurable in ξ and, for λ -a. e. ξ , completely monotone on R◦+ in u.

(5.16)

Using Bernstein’s theorem, this theorem is proved from Theorem 5.5 as Theorem 4.10 is from Theorem 4.9. In (4.5) of [42] kξ (u) is required not to be identically zero in u and to tend to zero as u → ∞, for λ -a. e. ξ . But it is not identically zero automatically from the definition of radial decomposition in Proposition 3.1; it tends to zero automatically from our definition of complete monotonicity in Section 1.5.


61

The following Theorem 5.8 and Proposition 5.9 are obtained in parallel to Theorem 4.15 and Proposition 4.16. Theorem 5.8 shows that, for 0 < α < 2, the class Re (Ψα ) ∩ {μ ∈ ID : Aμ = 0} is identical with the class of tempered α -stable distributions introduced by Rosiński [34]. He studied properties of the associated Lévy processes on Rd in detail. Theorem 5.8. Let −∞ < α < 2. Then μ ∈ Re (Ψα ) if and only if μ ∈ ID and νμ has a radial decomposition (λ (d ξ ), u−α −1kξ (u)du) satisfying (5.16). Proposition 5.9. Let 1 < α < 2. If μ ∈ Re (Ψα ), then (4.48) holds. Theorem 5.10. (i) Let −∞ < α < 1. Then R(Ψα ) = R0 (Ψα ) = Re (Ψα ). (ii) Let α = 1. Then μ ∈ R(Ψ1 ) if and only if μ ∈ Re (Ψ1 ) and μ has weak mean 0. (iii) Let α = 1. Then μ ∈ R0 (Ψ1 ) if and only if μ ∈ Re (Ψ1 ) and μ has weak mean 0 absolutely. (iv) Let 1 < α < 2. Then R(Ψα ) = R0 (Ψα ); μ ∈ R(Ψα ) if and only if μ ∈ Re (Ψα ) and μ has mean 0. Proof. We use Proposition 5.1. Assertions (i) and (iv) are proved similarly to Proposition 4.11 and Theorem 4.18 (iii). Proof of (ii). Method of the proof is the same as in Theorem 4.18 (ii). The “only if” part. Let μ ∈ R(Ψ1 ). Then μ = Ψ1 ρ for some ρ ∈ D(Ψ1 ). Define Iε as in (4.51) and Jε as Jε =

∞

ε

t −2 e−t dt

|tx|>1

txνρ (dx).

Then Iε = = = Jε = =

∞

t −2 e−t dt

0

S

S

S

S

ξ λρ (d ξ ) ξ λρ (d ξ ) ξ λρ (d ξ ) ξ λρ (d ξ )

1 0 we can check

t −1 e−t dt t −1 (e−t − 1(0,1)(t))dt,

62

K. Sato

1/(ε r) 1/r

For any fixed r > 0, I(ε , r) =

1/(ε r)

1/r

−

−

∞ ε ∨(1/r)

∞ ε ∨(1/r)

t −1 1(0,1) (t)dt = 0.

t −1 (e−t − 1(0,1)(t))dt → 0 as ε ↓ 0.

Now we can apply the dominated convergence theorem. Recall and use, for r ≥ 1, ∞ 0

t −1 |e−t − 1(0,1)(t)|dt =

1 0

t −1 (1 − e−t )dt +

∞ 1

|x|>1 |x|νρ (dx) < ∞

t −1 e−t dt < ∞

and, for 0 < r < 1 and 0 < ε < 1, 1/(ε r) 1/r

−

∞

|I(ε , r)| ≤

1/(ε r)

∞

=

ε ∨(1/r)

1/(ε r)

t −1 e−t dt ≤

1/r ∞

−

1/(ε r)

∞ 1/r

=−

∞ 1/(ε r)

,

e−t dt = e−1/(ε r) ≤ e−1/r .

Therefore Iε − Jε → 0 as ε ↓ 0. The rest of the proof is similar to that of the “only if ” part of Theorem 4.18 (ii). The “if” part is also similar. Proof of (iii). Method is the same as in the proof of Theorem 4.21. The “only ρ if” part. Let μ ∈ R0 (Ψ1 ) with μ = Ψ1 ρ , ρ ∈ D0 (Ψ1 ). Let (ν¯ ρ , λr ) be a spherical ρ decomposition of νρ such that λr , r ∈ R◦+ , are probability measures on S. Then

νμ (B) = = = =

∞ 0 ∞

t −2 e−t dt t −2 e−t dt

0

R◦+

∞

rν¯ ρ (dr) u−2 du

d

R◦+ ∞ 0

(0,∞)

0

1B (tx)νρ (dx)

R

ν¯ ρ (dr)

S

1B (trξ )λrρ (d ξ )

u−2 e−u/r du 1B (uξ )λrρ (d ξ ) S

re−u/r 1B (uξ )λrρ (d ξ ) ν¯ ρ (dr). S

Assuming that νρ = 0, define

λuμ (E) =

(0,∞)

re−u/r λrρ (E)ν¯ ρ (dr),

E ∈ B(S).

μ

μ

Then {λu : u ∈ R◦+ } is a measurable family of measures on S such that λu (S) < ∞ μ for a.e. u > 0. Moreover, λu (S) > 0, u ∈ R◦+ . We have

νμ (B) =

∞ 0

u−2 du

S

1B (uξ )λuμ (d ξ ),

B ∈ B(Rd \ {0}),


63

μ

μ

and, after redefining λu appropriately for u in a Lebesgue-null set, (u−2 du, λu (d ξ )) is a spherical decomposition of νμ . Let I=

∞ 1

μ u du uξ λu (d ξ ) , S −2

Then

∞

J=

t

e dt txνρ (dx) . |tx|>1

−2 −t

0

u−1 du re−u/r ν¯ ρ (dr) ξ λrρ (d ξ ) ◦ 1 R S + 1 t −1 dt re−1/(tr) ν¯ ρ (dr) ξ λrρ (d ξ ) . = ∞

I=

R◦+

0

S

Let

J =

1

t

−1

0

1 −1 ρ dt xνρ (dx) = t dt rν¯ ρ (dr) ξ λr (d ξ ) . 0 |tx|>1 (1/t,∞) S

Then J is finite if and only if J is finite, since ∞

t

∞ e dt xνρ (dx) ≤ t −1 e−t dt |x|νρ (dx) |x|>1/t |x|>1/t 1

−1 −t

1

≤

|x|≤1

|x|νρ (dr)

On the other hand, let I =

1 0

∞

t −1 dt

1/|x|

−t

e dt +

(1/t,∞)

|x|>1

|x|νρ (dr)

re−1/(tr) ν¯ ρ (dr)

S

∞ 1

e−t dt < ∞.

ξ λrρ (d ξ ) .

Then I is finite if and only if I is finite, since 1

t −1 dt

0

re−1/(tr) ν¯ ρ (dr) ≤

(0,1]

1

t −1 e−1/(2t) dt

0

(0,1]

re−1/(2r) ν¯ ρ (dr) < ∞

and 1

t −1 dt

0

(1,1/t]

re−1/(tr) ν¯ ρ (dr) = =

∞ 1

∞ 1

rν¯ ρ (dr) rν¯ ρ (dr)

1/r

t −1 e−1/(tr) dt

0

1 0

u−1 e−1/u du < ∞.

Finally we claim that I is finite if and only if J is finite. It is enough to show that 1

t 0

−1

−1/(tr) ¯ ¯ dt re νρ (dr) − rνρ (dr) < ∞. (1/t,∞) (1/t,∞)

64

K. Sato

This is proved to be true because 1 0

t −1 dt

r |e−1/(tr) − 1| ν¯ ρ (dr) =

(1/t,∞)

=

(1,∞)

(1,∞)

≤ const

rν¯ ρ (dr) rν¯ ρ (dr) (1,∞)

1 1/r r 1

(1 − e−1/(tr) )t −1 dt

(1 − e−1/u)u−1 du

rν¯ ρ (dr) < ∞,

since 1 − e−1/u = O(1/u) as u → ∞. The rest and the proof of the “if” part are a simple modification of the proof of Theorem 4.21. 0 , and K e Now let us express K∞,α , K∞, α ∞,α by the ranges of Ψα .

Theorem 5.11. Let −∞ < α < 2. Then K∞,α = R(Ψα ),

(5.17)

0 0 K∞, α = R (Ψα ),

(5.18)

= R (Ψα ).

(5.19)

e K∞, α

e

Proof. This follows from Theorems 4.15, 4.18, 4.21, 5.8, and 5.10.

Let us look at the ranges of Ψα as a family with parameter α . Proposition 5.12. For −∞ < α < 2

R(Ψα )

α ∈(α ,2)

R(Ψα ),

Re (Ψα )

α ∈(α ,2)

R0 (Ψα )

α ∈(α ,2)

R0 (Ψα ),

Re (Ψα ).

(5.20)

For 0 < α ≤ 2

R(Ψβ ) R(Ψα ),

β ∈(−∞,α )

R0 (Ψβ ) R0 (Ψα ),

β ∈(−∞,α )

R (Ψβ ) R (Ψα ). e

e

(5.21)

β ∈(−∞,α )

See Propositions 4.5, 4.8, 4.15, and 4.17 of [42]. Proposition 5.13. Let 0 < α < 2.

(i) If μ ∈ Re (Ψα ), then Rd |x|β μ (dx) < ∞ for all β ∈ (0, α ). (ii) There is μ ∈ R0 (Ψα ) such that Rd |x|α μ (dx) = ∞. (iii) There is μ ∈ R0 (Ψα ) which is not Gaussian and satisfies, for all α > 0, α Rd |x| μ (dx) < ∞. (iv) There is μ ∈ R0 (Ψ0 ) = T such that, for all α > 0, Rd |x|α μ (dx) = ∞.


65

For (i)–(iii), see the proof of Proposition 4.10 of [42]. For (iv), see Proposition 4.12 of [42]. The ranges of Ψα have the following relations with the classes Sα and S0α of α -stable and strictly α -stable distributions. Proposition 5.14. (i) Let 0 < α ≤ 1. We have

Sα ⊂

R0 (Ψβ ).

(5.22)

β ∈(0,α )

If μ ∈ Sα and μ is not a δ -measure, then μ ∈ Re (Ψα ). (ii) Let 1 < α ≤ 2. We have S0α ⊂ If μ ∈ Sα \ S0α , then μ ∈ then μ ∈ Re (Ψα ).

β ∈(0,α )

β ∈(1,2] R

R0 (Ψβ ).

0 (Ψ ). If β

(5.23)

μ ∈ Sα and μ is not a δ -measure,

See the proof of Proposition 4.7 of [42]. Remark 5.15. Open question: Do the limiting classes contain a distribution other than the Gaussian one?

R0 (Ψα ) and

α 0, r−a f (r) is monotone of order n on R◦+ . Proof. Apply Lemma 5.17 for f1 = f and f2 = r−a , which is completely monotone on R◦+ . Theorem 5.19. Let n be a positive integer. Let −∞ < α < α < 2. Then Kn,α Kn,α ,

Kn,0 α Kn,0 α ,

and Kn,e α Kn,e α .

(5.24)

Proof. Let us prove Kn,e α Kn,e α . Let μ ∈ Kn,α . By Theorem 4.15, νμ has a radial

decomposition (λ (d ξ ), u−α −1 kξ (u)du) such that kξ (u) is measurable in (ξ , u) and kξ (u) is monotone of order n on R◦+ in u. Notice that u−α −1 kξ (u) = u−α −1 kξ (u)

with kξ (u) = uα −α kξ (u). It follows from Lemma 5.18 that kξ (u) is monotone of order n on R◦+ . Hence μ ∈ Kn,α . Thus Kn,e α ⊃ Kn,e α . To show the strictness of the inclusion, let λ be a non-zero finite measure on S and let k(u) = (1 − u)n−11(0,1) (u), which is monotone of order n on R◦+ (Example 2.17 (a)). Let ν be the Lévy measure of polar product type (λ (d ξ ), u−α −1 k(u)du). Let μ ∈ ID with νμ = ν . Then μ ∈ Kn,e α . But μ ∈ Kn,e α , because the function k satisfying u−α −1 k(u) = u−α −1 k (u) is expressed as k (u) = uα −α (1 − u)n−11(0,1) (u), which is not monotone of any order on R◦+ by virtue of Proposition 2.13 (iv). Hence Kn,e α \ Kn,e α = 0. / The first and second relations in (5.24) are obtained from the third by the use of Theorems 4.18 and 4.21. Remark 5.20. Open question: Is Lemma 5.18 true for p ∈ R◦+ in place of n ? If the answer is affirmative, then Theorem 5.19 is true for p ∈ R◦+ in place of n. Iksanov, Jurek, and Schreiber [10] contains the identity

Φρ = (Φ¯ 1,−1 Φρ ) ∗ (Φ¯ 1,−1ρ ) = Φ¯ 1,−1 ((Φρ ) ∗ ρ ) for ρ ∈ D(Φ ).


67

This is generalized to the following identity for the family {Φ¯ 1,α : α ∈ (−∞, 2)}. Essentially the same result is given by Czy˙zewska-Jankowska and Jurek [7]. We use Propositions 3.19 and 3.20. Theorem 5.21. Let −∞ < α < α < 2. If ρ ∈ D0 (Φ¯ 1,α ), then ρ ∈ D0 (Φ¯ 1,α ), Φ¯ 1,α ρ ∈ D0 (Φ¯ 1,α ), and Φ¯ 1,α ρ = Φ¯ 1,α Φ¯ 1,α (ρ α −α ) ∗ Φ¯ 1,α ρ .

(5.25)

Proof. Recall that ∞ 0

|Cρ ( f¯1,α (s)z)|ds =

1 0

|Cρ (tz)|t −α −1 dt.

Suppose that ρ ∈ D0 (Φ¯ 1,α ) and let μ = Φ¯ 1,α ρ . Then 01 |Cρ (tz)|t −α −1 dt < ∞. Hence 01 |Cρ (tz)|t −α −1 dt < ∞, that is, ρ ∈ D0 (Φ¯ 1,α ). Further,

1 0

1 t −α −1 dt Cρ (stz)s−α −1 ds 0 0 t 1 = t α −α −1 dt Cρ (uz)u−α −1 du

|Cμ (tz)|t −α −1 dt =

1

0

≤ =

0

1

t

α −α −1

t

dt

0

1 0

0

|Cρ (uz)|u−α −1 du

|Cρ (uz)|u−α −1 du

−1

≤ (α − α )

1 0

1

t α −α −1 dt

u

|Cρ (uz)|u−α −1 du < ∞.

= Φ¯ 1,α μ . Then Hence μ ∈ D0 (Φ¯ 1,α ). Let μ Cμ (z) = =

1 0

1

Cμ (tz)t −α −1 dt =

t α −α −1 dt

0

= (α − α )−1

t

1 0

0

1

t −α −1 dt

0

1 0

Cρ (uz)u−α −1 du =

Cρ (uz)(u

−α −1

Cρ (stz)s−α −1 ds 1 0

Cρ (uz)u−α −1 du

1

t α −α −1 dt

u

− u−α −1)du.

Hence (α − α )Cμ (z) + which is (5.25).

1 0

Cρ (uz)u−α −1 du =

1 0

Cρ (uz)u−α −1 du,

The fact K1,0 α ⊃ K1,0 α for −∞ < α < α < 2 follows also from the theorem above.

68

K. Sato

Maejima, Matsui, and Suzuki [21] and Maejima and Ueda [28] studied essentially the same class as K1,α with parameter α . They gave the description of the triplet of μ ∈ K1,α and a kind of decomposability which generalizes (1.2), and introduced a generalization of Ornstein–Uhlenbeck type process which corresponds to this class. An earlier paper [14] of Jurek is also related.

6 Second Two-Parameter Extension Lp,α of the Class L of Selfdecomposable Distributions 6.1 Λ p,α and Λ Lp,α For −∞ < α < ∞ and p > 0 we have introduced j p,α (t), l p,α (s), and Λ p,α in Section 1.6. Namely, j p,α (t), 0 < t ≤ 1 is defined by (1.14); b p,α = j p,α (0+) equals (−α )−p for α < 0 and ∞ for α ≥ 0; t = l p,α (s), 0 ≤ s < b p,α , if and only if s = j p,α (t), 0 < t ≤ 1; l p,α (s) is defined to be zero if α < 0 and s ≥ b p,α ; L of L´ Λ p,α = Φ f with f = l p,α in (3.24). Define the transformation Λ p, evy measures α L = Φ L in Definition 3.25 with f = l as Λ p, . p,α α f We note the following special cases. If p = 1, then j1,α (t) = g¯1,α (t),

l1,α (s) = f¯1,α (s),

(6.1)

so that the explicit forms are given in (4.16)–(4.18). Thus Λ1,α = Φ¯ 1,α and Λ1,0 = Φ . If p > 0 and α = 0, then j p,0 (t) = c p+1 (− logt) p ,

0 ≤ t ≤ 1,

l p,0 (s) = exp(−(Γp+1s)1/p ),

(6.2)

s ≥ 0.

(6.3)

From the definition of absolute definability we have

ρ ∈ D0 (Λ p,α ) It follows that

⇔

1 0

|Cρ (tz)|(− logt) p−1t −α −1dt < ∞.

D0 (Λ p,α ) ⊃ D0 (Λ p ,α ) if 0 0, then, as s → ∞, l p,α (s) ∼ (αΓp s)−1/α (α −1 log s)(p−1)/α

for p > 0.

Proof. Let α > 0. We have j p,α (t) = α −1 c p (− logt) p−1t −α (1 + o(1)),

t ↓ 0.

(6.5)


69

Let s = j p,α (t) and t = l p,α (s) = l(s) = s−1/α l (s). Then s = α −1 c p (− log l(s)) p−1 l(s)−α (1 + o(1)),

s → ∞.

(6.6)

If p = 1, this shows (6.5). Assume p = 1 in the following. It follows from (6.6) that l (s)α /(p−1) = (α −1 c p )1/(p−1)(α −1 log s − logl (s))(1 + o(1)). Define l (s) as l (s) = (α −1 c p )1/α (α −1 log s)(p−1)/α l (s). Then we see that

(p − 1) loglog s α log l (s) B l (s)α /(p−1) = 1 − − − (1 + o(1)), log s log s log s

(6.7)

where B is a constant independent of s. Let sn → ∞ be a sequence such that l (sn ) tends to some C ∈ [0, ∞]. If C is 0 or ∞, then we have a contradiction from (6.7) when p > 1 as well as when p < 1. Hence C = 0, ∞. Then we obtain C = 1 again from (6.7). It follows that l (s) → 1 as s → ∞, which shows (6.5). L is as follows: Theorem 6.2. Let −∞ < α < ∞ and p > 0. The domain of Λ p, α L L D(Λ p, α) = M L D(Λ p,0 ) L D(Λ p,α )

if α < 0,

= {ν ∈ M : L

= {ν ∈ M : L

(6.8)

|x|>2 (log |x|) |x|>2

p

ν (dx) < ∞} if α = 0,

(log |x|) p−1 |x|α ν (dx)

< ∞}

if 0 < α < 2, L D(Λ p, α ) = {δ0 }

(6.9)

(6.10)

if α ≥ 2.

(6.11)

Recall that Λ1,L α = Φ¯ 1,L α and notice that this theorem for p = 1 is consistent with Theorem 4.1. Proof of Theorem 6.2. Let 0 < α < 2. Given ν , we express the measure ν in Definition 3.25 for f = l p,α as

ν (B) = c p

1 0

(− logt) p−1t −α −1dt

Rd

1B (tx)ν (dx)

(6.12)

for B ∈ B(Rd \ {0}). We use the fact that u 0

1 u

(− logt) p−1t q dt ∼ (q + 1)−1(− log u) p−1 uq+1 ,

u ↓ 0,

for p ∈ R, q > −1, (− logt) p−1t q dt ∼ (−q − 1)−1(− log u) p−1 uq+1,

(6.13)

u ↓ 0,

for p > 0, q < −1.

(6.14)

70

K. Sato

Thus |x|≤1

|x|2 ν (dx) = c p = cp ≤ C1

|x|>1

ν (dx) = c p = cp

1 0


Rd

|x|2 ν (dx)

|x|≤2 1 0

0

|x|2 ν (dx) + C2

|x|>1

ν (dx)

12


≤ C3

1∧(1/|x|)

(log |x|) p−1 |x|α ν (dx),

|tx|>1

ν (dx)

(− logt) p−1t −α −1 dt

ν (dx) + C4

|x|>2

(log |x|) p−1 |x|α ν (dx).

Here C1 , . . . ,C4 are positive constants. Similarly we can show the reverse estimates. Hence (6.10) is true. If α < 0, then (6.8) comes from b p,α < ∞. If α = 0, then we have (6.9) as in Theorem 5.15 of [41] and Proposition 4.3 of [43]. If α ≥ 2, then we have (6.11) by a similar argument. Let us study the domains of Λ p,α . Theorem 6.3. Let −∞ < α < ∞ and p > 0. (i) If α < 0, then D0 (Λ p,α ) = D(Λ p,α ) = De (Λ p,α ) = ID. (ii) If 0 ≤ α < 2, then L De (Λ p,α ) = {ρ ∈ ID : νρ ∈ D(Λ p, α )}.

(iii) If 0 ≤ α < 1, then D0 (Λ p,α ) = D(Λ p,α ) = De (Λ p,α ). (iv) If α = 1 and p ≥ 1, then D0 (Λ p,1 ) D(Λ p,1 ) De (Λ p,1 ), L D(Λ p,1 ) = {ρ ∈ ID : νρ ∈ D(Λ p,1 ),

lim

a→∞ |x|>1

Rd

xρ (dx) = 0,

x(log(|x| ∧ a)) p νρ (dx) exists in Rd },

(6.15)


71

L D0 (Λ p,1 ) = {ρ ∈ ID : νρ ∈ D(Λ p,1 ), xρ (dx) = 0, d R 1 (− logt) p−1t −1 dt xνρ (dx) < ∞}. |x|>1/t 0

(6.16)

(v) If 1 < α < 2, then L D0 (Λ p,α ) = D(Λ p,α ) = {ρ ∈ ID : νρ ∈ D(Λ p, α ),

Rd

xρ (dx) = 0}

D (Λ p,α ). e

(vi) If α ≥ 2, then D0 (Λ p,α ) = D(Λ p,α ) = {δ0 } De (Λ p,α ) = {δγ : γ ∈ Rd }.

Proof. If 1 < α < 2 or if α = 1 with p ≥ 1, then Rd |x|ρ (dx) < ∞ for ρ satisfying L ) (see Theorem 6.2). We write l(s) = l νρ ∈ D(Λ p, p,α (s) for simplicity. We use α C1 ,C2 , . . . for positive constants. To show (i), note that b p,α < ∞ for α < 0. To show 6.1. (ii), note that 0∞ l(s)2 ds < ∞ for 0 ≤ α < 2 by (6.5) of Proposition Let us prove (iii). Let 0 < α < 1. Let ρ ∈ De (Λ p,α ). We have 0∞ l(s)ds < ∞ from (6.5). Choosing s0 = j p,α (t0 ) > 0 such that l(s) < 1 for s > s0 , we have ∞

|x(1{|l(s)x|≤1} − 1{|x|≤1})|νρ (dx) ∞ = l(s)ds |x|1{|l(s)x|>1} νρ (dx) +

l(s)ds s0

Rd

s

=

0∞

l(s)ds s0

= cp = cp = C1

t0

0

|x|≤1

|x|>1

|x|>1

|x|1{|l(s)x|≤1} νρ (dx)

(− logt) p−1t −α dt

|x|>1

|x|νρ (dx)

|x|>1

= C2 + C3

|x|>1/t0

|x|>1

t0 ∧(1/|x|) 0

|x|νρ (dx) + c p

|x|1{|l(s)x|≤1} νρ (dx)

|x|1{|tx|≤1} νρ (dx)


|x|>1/t0

|x|νρ (dx)

1/|x| 0


(log |x|) p−1 |x|α νρ (dx) < ∞

by (6.13) and (6.10). Hence it follows from Proposition 3.18 (iii) that ρ ∈ D0 (Λ p,α ). Thus De (Λ p,α ) ⊂ D0 (Λ p,α ) and the assertion is true. In the case α = 0, the argument is similar; it is done in Theorem 5.15 of [41] and Proposition 4.3 of [43]. Proof of (iv) is as follows. Let α = 1 and p ≥ 1. Assume that ρ ∈ D(Λ p,1 ). Then γμt given by

t γμt = l(s)ds γρ + x(1{|l(s)x|≤1} − 1{|x|≤1}) νρ (dx) 0

Rd

72

K. Sato

is convergent in Rd as t → ∞ (Proposition 3.18). Since

x(1{|l(s)x|≤1} − 1{|x|≤1} ) νρ (dx) →

Rd

xνρ (dx),

|x|>1

and since 0∞ l(s)ds = ∞ from (6.5), we have γρ = − mean 0. Hence we have, with ε = l(t),

γμt =

t

l(s)ds 0

= −c p = −c p

1

ε

(− logt) p−1t −1 dt

|x|>1

= −c p+1

Rd

x(1{|l(s)x|≤1} − 1)νρ (dx) = −

xνρ (dx)

|x|>1

1

|tx|>1

ε ∨(1/|x|)

s → ∞,

|x|>1 xνρ (dx),

t

l(s)ds 0

that is, ρ has

|l(s)x|>1

xνρ (dx)

xνρ (dx)

(− logt) p−1t −1 dt

x(log(|x| ∧ (1/ε ))) p νρ (dx).

Therefore ρ is in the right-hand side of (6.15). Conversely, if ρ is in the righthand side of (6.15), then it follows from the equalities above that γμt is convergent, hence ρ ∈ D(Λ p,1 ). To see (6.16), assume that ρ ∈ D0 (Λ p,1 ). Then ρ ∈ D(Λ p,1 ), L νρ ∈ D(Λ p,1 ), and ρ has mean 0. We have ∞> l(s)ds γρ + x(1{|l(s)x|≤1} − 1{|x|≤1}) νρ (dx) d R 0 ∞ 1 p−1 −1 = l(s)ds xνρ (dx) = c p (− logt) t dt ∞

|l(s)x|>1

0

|tx|>1

0

xνρ (dx) ,

and ρ is in the right-hand side of (6.16). These equalities also show the converse. Let us show (v). Let 1 1 xνρ (dx), that is, ρ has mean 0. If ρ ∈ ID has mean L ), then ρ ∈ D0 (Λ 0 and νρ ∈ D(Λ p, p,α ), since α ∞ 0

l(s)ds γρ + x(1{|l(s)x|≤1} − 1{|x|≤1}) νρ (dx) d R ∞ 1 p−1 −α l(s)ds xνρ (dx) = c p (− logt) t dt = 0

≤ cp ≤ C4

|l(s)x|>1

|x|>1

|x|>1

|x|νρ (dx)

1 1/|x|

0

|tx|>1

xνρ (dx)


(log |x|) p−1 |x|α νρ (dx) < ∞

from (6.14) and (6.10). ∞ 2 ∞ To see assertion (vi) for α ≥ 2, notice that we have 1 l(s) ds = ∞ as well as 1 l(s)ds = ∞ from (6.5). Combining this with (6.11), we obtain the result.


73

Remark 6.4. Open problem: Describe the domains of Λ p,1 for 0 < p < 1. Remark 6.5. Consider the case where α = 1 and p ≥ 1. In this case, L ), D(Λ p,1 ) {ρ ∈ ID : νρ ∈ D(Λ p,1

Rd

xρ (dx) = 0}.

(6.17)

Indeed, let λ = δξ0 , ξ0 ∈ S, q ∈ (p, p + 1], and

ν (B) =

S

λ (d ξ )

∞

1B (rξ )r−2 (log r)−q dr.

2

L ) by Theorem 6.2. Let ρ ∈ ID be Then |x|>2 |x|ν (dx) < ∞. Since q > p, ν ∈ D(Λ p,1 such that Aρ arbitrary, νρ = ν , and γρ = − |x|>1 xν (dx). Then ρ is in the right-hand side of (6.17), but ρ ∈ D(Λ p,1 ) by virtue of (6.15), because

|x|>1

x(log(|x| ∧ (1/ε ))) pνρ (dx) = p

1 ε

= pξ0 =

(− logt) p−1 t −1 dt

1 ε

pξ0 q−1

|x|>1/t

(− logt) p−1 t −1 dt

1 ε

xνρ (dx)

∞ (1/t)∨2

(log r)−q r−1 dr

(− logt) p−1 t −1 (− log(t ∧ (1/2)))1−q dt,

which tends to the infinity point in the direction of ξ0 as ε ↓ 0, since q ≤ p + 1.

6.2 Range of Λ Lp,α Theorem 6.6. Let −∞ < α < 2 and p > 0. L ) with a radial decomposition (λ (d ξ ), ν ) and let ν L ν.

= Λ p, (i) Let ν ∈ D(Λ p, ξ α α Then ν has a radial decomposition (λ (d ξ ), u−α −1hξ (log u)du), where

hξ (y) = c p

νξ (E) = L (ii) Λ p, α is one-to-one.

(y,∞)

(0,∞)

(w − y) p−1eα w νξ (dw),

1E (log r)νξ (dr),

y ∈ R,

E ∈ B(R).

(6.18) (6.19)

74

K. Sato

Proof. To show (i), notice that it follows from (6.12) that

ν (B) = c p = cp = cp =

S

S

S

S

λ (d ξ ) λ (d ξ ) λ (d ξ )

λ (d ξ )

(0,∞)

νξ (dr)

0

rα νξ (dr)

(0,∞) ∞ −α −1

u

0

(0,∞)

1

(− logt) p−1t −α −1 1B (trξ )dt

r 0

(log(r/u)) p−1 u−α −1 1B (uξ )du

1B (uξ )du

(u,∞)

(log(r/u)) p−1 rα νξ (dr)

1B (uξ )u−α −1hξ (log u)du,

where hξ is defined by (6.18) and (6.19). Proof of (ii) is as follows. Similarly to the proof of Theorem 4.9 (ii), we see that there is a measurable function c(ξ ) satisfying 0 < c(ξ ) < ∞, c(ξ )λ (d ξ ) = λ (d ξ ), and u−α −1 hξ (log u)du = c(ξ )u−α −1 hξ (log u)du on R◦+ for λ -a. e. ξ . Thus hξ (y)dy = c(ξ )hξ (y)dy on R for λ -a. e. ξ . For λ -a. e. ξ , hξ (log u)du and hξ (log u)du are locally finite measures on R◦+ , hence hξ (y)dy and hξ (y)dy are locally finite measures on R, and also eα w νξ (dw) and eα w νξ (dw) are locally finite measures on R. Now from Theorem 2.10 on the one-to-one property of I p we obtain eα w νξ (dw) = c(ξ )eα w νξ (dw) for λ -a. e. ξ . Hence νξ = c(ξ )νξ for λ -a. e. ξ . This shows that ν = ν .

L Theorem 6.7. Let −∞ < α < 2 and p > 0. A measure η on Rd belongs to R(Λ p, α ) if L − α −1 and only if η is in M and has a radial decomposition (λ (d ξ ), u hξ (log u)du) such that

hξ (y) is measurable in (ξ , y) and, for λ -a. e. ξ , monotone of order p on R in y.

(6.20)

Proof. This follows from Theorem 6.6. We can supply the details, modifying the proof of Theorem 4.10. Notice that monotonicity of order p on R◦+ and on R appears in (4.31) and in (6.20), respectively. This is why we have studied in Section 2 monotonicity on R◦+ and R both.

6.3 Classes Lp,α , L0p,α , and Lep,α Define, for −∞ < α < 2 and p > 0, L p,α = L p,α (Rd ) = R(Λ p,α ),

(6.21)

L0p,α = L0p,α (Rd ) = R0 (Λ p,α ),

(6.22)

= R (Λ p,α ).

(6.23)

Lep,α

= Lep,α (Rd )

e


75

The notation Ln,0 for positive integers n is already introduced in Section 1.2 as the classes of n times selfdecomposable distributions, but this is consistent with (6.21) for p = n and α = 0, because the known characterization of the Lévy measures of n times selfdecomposable distributions in Theorem 3.2 of Sato [37] coincides with the description of L p,α in Theorem 6.12. Another proof is to use the expression (6.3) for ln,0 and to recall the result mentioned in Section 1.2. A third proof is to use Λ p+q,0 = Λq,0Λ p,0 to be shown in Theorem 7.3 (ii). Proposition 6.8. We have L0p,α = L p,α = Lep,α L0p,1 L0p,α

⊂ L p,1 ⊂ Lep,1 , = L p,α ⊂ Lep,α

for −∞ < α < 1,

(6.24) (6.25)

for 1 < α < 2.

(6.26)

Proof. This is parallel to Proposition 4.11. For 0 ≤ α < 1, (6.24) is proved as in Proposition 4.5, using Theorem 6.3 instead of Theorem 4.2. For 1 ≤ α < 2, use Theorem 6.3 and (3.28). Theorem 6.9. Let −∞ < α < 2 and p > 0. Then μ ∈ Lep,α if and only if μ ∈ ID and its Lévy measure νμ has a radial decomposition (λ (d ξ ), u−α −1 hξ (log u)du) satisfying (6.20).

Proof. Use Proposition 3.27 and Theorem 6.7. Proposition 6.10. Let 0 < α < 2, p > 0, and μ ∈ Lep,α . Then Rd

|x|β μ (dx) < ∞ for all β ∈ (0, α ).

(6.27)

L ν for some ν ∈ D(Λ L ) and Proof. We have νμ = Λ p, p,α α

|x|>1

|x|β νμ (dx) = c p = cp

1 0

(− logt) p−1t −α −1 dt

|x|>1

≤ const

|x|β ν (dx)

|x|>1

1 1/|x|

|tx|>1

|tx|β ν (dx)

(− logt) p−1t β −α −1dt

(log |x|) p−1 |x|α ν (dx) < ∞

by (6.13) and Theorem 6.2. Remark 6.11. Let 0 < α < 2 and p > 0.

(i) There is μ ∈ Lep,α such that Rd |x|α μ (dx) = ∞. (ii) There is a non-Gaussian μ ∈ Lep,α (Rd ) such that Rd |x|α μ (dx) < ∞ for all α > 0.

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K. Sato

Indeed, (i) is a consequence of Theorem 7.11 and Proposition 7.16 in the later section. To see (ii), choose h(y) = (−y) p−1 1(−∞,0)(y) and consider μ such that νμ has a radial decomposition (λ , u−α −1 h(log u)du) = (λ , u−α −1 (− log u) p−11(0,1) (u)du) with a nonzero finite measure λ . We give characterization of L p,α for α = 1. Recall that L p,α = L0p,α if α = 1. Theorem 6.12. Let μ ∈ ID. (i) Let −∞ < α < 1 and p > 0. Then μ ∈ L p,α if and only if νμ has a radial decomposition (λ (d ξ ), u−α −1 hξ (log u)du) satisfying (6.20). (ii) Let 1 < α < 2 and p > 0. Then μ ∈ L p,α if and only if νμ has a radial decomposition (λ (d ξ ), u−α −1 hξ (log u)du) satisfying (6.20) and μ has mean 0. Proof. Assertion (i) is from Proposition 6.8 and Theorem 6.9. Let us prove assertion (ii). Let μ ∈ L p,α . Then μ ∈ Lep,α from (6.26), and Theorem 6.9 says that νμ has (λ (d ξ ), u−α −1hξ (log u)du) satisfying (6.20). We have μ = Λ p,α ρ for some ρ ∈ D(Λ p,α ). Thus, by Theorems 6.2 and 6.3, |x|>2 (log |x|) p−1 |x|α νρ (dx) < ∞ and Rd xρ (dx) = 0. Hence γρ = − |x|>1 xνρ (dx). Let l = l p,α . It follows from Proposition 5.9 that ∞

ds 0

since this integral equals

|l(s)x|>1

|l(s)x|νρ (dx) < ∞,

|x|>1 |x|νμ (dx).

γμ = −

∞

It follows that

ds 0

(6.28)

|l(s)x|>1

l(s)xνρ (dx),

(6.29)

and hence γμ = − |x|>1 xνμ (dx), that is, μ has mean 0. Conversely, assume that νμ has the property stated and that μ has mean 0. Then by Theorem 6.9, μ ∈ Lep,α and L ). Choose ν such that Λ L ν = ν . Then (6.28) and (6.29) hold with ν νμ ∈ R(Λ p, μ p,α α in place of νρ . Let A = ( 0∞ l p,α (s)2 ds)−1 Aμ and γ = − |x|>1 xν (dx). Then ρ ∈ ID with triplet (A, ν , γ ) belongs to D(Λ p,α ) from Theorem 6.3 and we have Λ p,α ρ = μ . Remark 6.13. Open problem: Describe the classes L p,1 (Rd ) and L0p,1 (Rd ) for p > 0. Theorem 6.14. Let −∞ < α < 2 and 0 < p < ∞. The mapping Λ p,α is one-to-one. This is proved from Theorem 6.6 (ii) in the same way as Theorem 4.23. Here is the continuity property of distributions in L p,α . Theorem 6.15. (i) Let μ be a nondegenerate distribution in Lep,α with 0 ≤ α < 2 and p > 0. Then μ is absolutely continuous with respect to d-dimensional Lebesgue measure.


77

(ii) Let μ = Λ p,α ρ with α < 0 and p > 0. Then νμ is a finite measure if and only if νρ is a finite measure. In particular, for any α < 0 and p > 0, L p,α contains some compound Poisson distribution. Proof. This is proved by the same idea as Theorem 4.24. The key formulas are, for ν 0 and ρ similarly defined, ∞ 0

u−α −1 hξ (log u)du = c p

for α ≥ 0 and

(0,∞)

rα νξ0 (dr)

r 0

u−α −1(log(r/u)) p−1du = ∞

νμ (Rd ) = (−α )−p νρ (Rd )

for α < 0.

6.4 Relation Between Kp,α and Lp,α We have K1,α = L1,α , K1,0 α = L01,α , and K1,e α = Le1,α for −∞ < α < 2 and, in particular, K1,0 = L1,0 = L. See (1.18), (1.19), and (6.1). Theorem 6.16. Let n be an integer ≥ 2. Then Kn,e α Len,α

for −∞ < α < 2

(6.30)

Kn,α Ln,α

for α ∈ (−∞, 1) ∪ (1, 2),

(6.31)

L0n,α

for α ∈ (−∞, 1) ∪ (1, 2).

(6.32)

Kn,0 α

Proof. To see Kn,e α ⊃ Len,α , compare Theorems 4.15 and 6.9; we can see that it is enough to show that if h(y) is a function monotone of order n on R, then h(log u) is monotone of order n on R◦+ . Let us prove this assertion. If n = 1, then the assertion is clear from Proposition 2.11 (i). Let n ≥ 2 and assume that the assertionis true for n − 1 in place of n. Let h(y) be monotone of order n on R. Then h(y) = y∞ ϕ (s)ds with ϕ monotone of order n − 1 on R, ∞

h(log u) =

log u

ϕ (s)ds =

∞ u

ϕ (logt)t −1 dt,

and ϕ (logt) is monotone of order n − 1 on R◦+ . Since t −1 is completely monotone on R◦+ , ϕ (logt)t −1 is monotone of order n − 1 on R◦+ by Lemma 5.18. Thus h(log u) is monotone of order n on R◦+ . Hence Kn,e α ⊃ Len,α . Next, let us show that Kn,e α \ Len,α = 0/ for n ≥ 2. Let k(u) = (1 − u)

n−1

1(0,1)(u) =

(u,∞)

(s − u)n−1 δ1 (ds),

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K. Sato

which is monotone of order n on R◦+ . Let h(y) = k(ey ) = (1 − ey )n−1 1(−∞,0) (y). Then h(y) is not monotone of order n on R, since h (y) = (n − 1)(n − 2)(1 − ey)n−3 e2y − (n − 1)(1 − ey)n−2 ey < 0 for y sufficiently close to −∞. Hence, for any finite measure λ on S, the Lévy measure with radial decomposition (λ , u−α −1 k(u)du) belongs to R(Φ¯ n,L α ) \ R(Λn,L α ). For α ∈ (−∞, 1), (6.31) and (6.32) follow from (6.30) by the equalities (4.35) and (6.24). For α ∈ (1, 2), (6.31) and (6.32) follow from (6.30) by adding the condition of having mean 0 in Theorems 4.18 and 6.12. 0 L0 for Remark 6.17. Open questions: (i) Is it true that Kn,1 Ln,1 and Kn,1 n,1 integers n ≥ 2? (ii) What is the relation between K p,α and L p,α for non-integer p > 0?

7 One-Parameter Subfamilies of {Lp,α } 7.1 Lp,α , L0p,α , and Lep,α for p ∈ (0, ∞) with fixed α We give a basic relation. Theorem 7.1. Let −∞ < α < 2, p > 0, and q > 0. Then L L Λq,L α Λ p, α = Λ p+q,α .

(7.1)

Proof. First note that a special case of (2.4) with σ = δ0 gives 0

c p cq

u

(−r)q−1 (r − u) p−1 dr = c p+q (−u) p+q−1,

u < 0,

that is, for 0 < w < 1, 1

c p cq

w

(− log u)q−1 (− log(w/u)) p−1 u−1 du = c p+q (− log w) p+q−1 .

Given ν ∈ ML (Rd ), let ν ( j) ({0}) = 0, j = 1, 2, and

ν (1) (B) = ν (2) (B) =

∞

ds 0 ∞

ds 0

R

d

Rd

1B (l p,α (s)x)ν (dx), 1B (lq,α (s)x)ν (1) (dx)

(7.2)


79

for B ∈ B(Rd \ {0}). Then ν (2) (B) = cq

1 0

= cq c p = cq c p = cq c p = c p+q

(− log u)q−1 u−α −1 du

1 0

1 0

Rd

(− log u)q−1 u−α −1 du (− log u)q−1 u−1 du

Rd 1 0

ν (dx)

1 0

1B (ux)ν (1) (dx)

1

u 0

0

(− logt) p−1 t −α −1 dt

Rd

(− log(w/u)) p−1 w−α −1 dw

1B (wx)w−α −1 dw

(− log w) p+q−1 w−α −1 dw

1

Rd

w

1B (utx)ν (dx) Rd

1B (wx)ν (dx)

(− log u)q−1 (− log(w/u)) p−1 u−1 du

1B (wx)ν (dx),

using (7.2). Hence

On the other hand,

ν (2) ∈ ML

⇔

L ν ∈ D(Λ p+q, α ).

ν (2) ∈ ML

⇔

ν (1) ∈ D(Λq,L α ).

Hence L ν ∈ D(Λ p+q, α)

⇔

L L (1) ν (1) ∈ D(Λq,L α ), ν ∈ D(Λ p, α ), Λ p,α ν = ν .

L L L L It follows that D(Λ p+q, α ) = D(Λq,α Λ p,α ) and that, if ν ∈ D(Λ p+q,α ), then L L L Λ p+q, α ν = Λq,α Λ p,α ν .

Corollary 7.2. We have L L R(Λ p, α ) ⊃ R(Λ p ,α ) for −∞ < α < 2 and 0 0, and q > 0. (i) If ρ ∈ D0 (Λ p+q,α ), then ρ ∈ D0 (Λ p,α ), Λ p,α ρ ∈ D0 (Λq,α ), and

(ii) If α = 1, then

Λ p+q,α ρ = Λq,α Λ p,α ρ

(7.4)

Λ p+q,α = Λq,α Λ p,α

(7.5)

Proof. Let us prove (i). Let ρ ∈ D0 (Λ p+q,α ). As in the proof of Theorem 7.1, 1

c p cq

0

q−1 −α −1

(− log u)

= c p+q

1 0

u

1

du 0

|Cρ (tuz)|(− logt) p−1t −α −1 dt

|Cρ (wz)|(− log w) p+q−1 w−α −1 dw,

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K. Sato

which is finite since ρ ∈ D0 (Λ p+q,α ). Then, we can use Fubini’s theorem and obtain 1

c p cq

0

(− log u)q−1 u−α −1du

= c p+q

1 0

1 0

Cρ (tuz)(− logt) p−1t −α −1 dt

Cρ (wz)(− log w) p+q−1 w−α −1 dw.

We have ρ ∈ D0 (Λ p,α ) from (6.4), and 1

cq

0

|CΛ p,α ρ (uz)|(− log u)q−1 u−α −1 du

≤ c p cq

1 0

(− log u)q−1 u−α −1du

1 0

|Cρ (tuz)|(− logt) p−1t −α −1dt < ∞.

Hence Λ p,α ρ ∈ D0 (Λq,α ) and (7.4) holds. To show (ii), let α = 1. Then we have D(Λr,α ) = D0 (Λr,α ) for all r > 0 by Theorem 6.3. If ρ ∈ D(Λ p+q,α ), then ρ ∈ D(Λq,α Λ p,α ) and Λq,α Λ p,α ρ = Λ p+q,α ρ by (i). It remains to show that D(Λq,α Λ p,α ) ⊂ D(Λ p+q,α ). Let ρ ∈ D(Λq,α Λ p,α ). L ) and This means that ρ ∈ D(Λ p,α ) and Λ p,α ρ ∈ D(Λq,α ). Hence νρ ∈ D(Λ p, α L ν ∈ D(Λ L ). Hence we have ν ∈ D(Λ L Λ p, ρ α ρ q,α p+q,α ) from Theorem 7.1. Hence ρ ∈ if α < 1, then ρ ∈ D(Λ p+q,α ) since De (Λ p+q,α ) = D(Λ p+q,α ). De (Λ p+q,α ). Now, If α > 1, then Rd xρ (dx) = 0 from ρ ∈ D(Λ p,α ), using Theorem 6.3, and hence ρ ∈ D(Λ p+q,α ). Corollary 7.4. For any positive integer n and α ∈ (−∞, 1) ∪ (1, 2), we have

Λn,0 = Φ · · · Φ n

and Λn,α = Φ¯ 1,α · · · Φ¯ 1,α , n

where Φ is defined by (1.11). Proof. Combine (7.5) with Λ1,0 = Φ and Λ1,α = Φ¯ 1,α .

Remark 7.5. Open question: Is (7.5) also true for α = 1? Corollary 7.6. For −∞ < α < 2 and 0 0 L0p,α

p ∈(p,∞)

L0p ,α

and Lep,α

p ∈(p,∞)

Lep ,α .

(7.7)


81

Proof. Let λ be a nonzero finite measure on S and let h(y) = (−y) p−1 1(−∞,0) (y). Then h(log u) = (− log u) p−11(0,1) (u). The measure ν of polar product type (λ (d ξ ), L )\ L u−α −1 h(log u)du) belongs to R(Λ p, p >p R(Λ p ,α ), since hξ (y) is monotone of α / order p but not of order p (Example 2.17 (a)). It follows that Lep,α \ p >p Lep ,α = 0. 0 0 If α < 1, then this also says that L p,α \ p >p L p ,α = 0. / If 1 < α < 2, then let μ ∈ ID be such that νμ = ν and, recalling that |x|>1 |x|ν (dx) < ∞, choose γμ = − |x|>1 xν (dx) to see that μ ∈ L0p,α \ p >p L0p ,α by Theorem 6.12. Assuming that L ν = ν and α = 1, let λ satisfy S ξ λ (d ξ ) = 0 and let ρ ∈ ID be such that Λ p,1 ρ γρ = 0. We consider the proof of Theorem 6.7 and see that νρ can be chosen to be of polar product type with the same λ . Hence Rd

x(1{|l p,1 (s)x|≤1} − 1{|x|≤1})νρ (dx) = 0,

which shows that ρ ∈ D0 (Λ p,1 ) by Proposition 3.18. Thus μ = Λ p,1 ρ has νμ = ν and γμ = 0 and belongs to L0p,1 \ p >p L0p ,1 . Remark 7.8. Since L p,α = L0p,α for α ∈ (−∞, 1) ∪ (1, 2), L p,α has the properties similar to Corollary 7.6 and Theorem 7.7 if α = 1. Open question: Is it true that L p,1 ⊃ L p ,1 for 0 p L p ,1 for p > 0? If α ≤ 0, then the class L0p,α is continuous for decreasing p in the following sense. Theorem 7.9. Let −∞ < α ≤ 0 and p > 0. Then

L0q,α = L0p,α .

(7.8)

q∈(0,p)

Proof. Let μ ∈ q∈(0,p) L0q,α . It is enough to prove that μ ∈ L0p,α . Let (λ (d ξ ), u−α −1 hξ (log u)du) be a radial decomposition of νμ . For any q ∈ (0, p) there is σξq ∈ D(I q ) such that hξ (y) = cq

(s − y)q−1σξ (ds), q

(y,∞)

y ∈ R.

Fix ξ for the moment and omit the subscript ξ . For −∞ < a < b < ∞ b a

h(y)dy =

b a

(I q σ q )(dy) ≥ cq+1

(a,b]

(s − a)q σ q (ds),

as in the proof of Proposition 2.1. Hence b a−1

h(y)dy ≥ cq+1

[a,b]

(s − a + 1)qσ q (ds) ≥ cq+1 σ q ([a, b]).

82

K. Sato

Hence by the diagonal method we can select a sequence qn ↑ p such that σ qn converges vaguely to a locally finite measure σ p on R, that is,

f (s)σ qn (ds) →

R

R

f (s)σ p (ds),

n→∞

for any continuous function f on R with compact support. We claim that σ p ∈ M∞p (R). We have, for 0 < β < q < p, ∞> =

∞ 1

u−α −1 h(log u)du =

∞ 0

dy cq

= cq+1

(0,∞)

e−α y h(y)dy ≥

0

σ (ds) =

q−1 q

(y,∞)

(s − y)

∞

sq σ q (ds) ≥ cq+1

(1,∞)

∞

h(y)dy 0

σ (ds)cq q

(0,∞)

s 0

(s − y)q−1dy

sβ σ q (ds).

Thus, for any continuous function g(s) ≤ sβ with compact support in (1, ∞), ∞ 1

u−α −1h(log u)du ≥ cq+1

(1,∞)

g(s)σ q (ds).

Letting q = qn and n → ∞, we get the same inequality with σ p in place of σ q . Hence ∞ 1

u−α −1 h(log u)du ≥ c p+1

(1,∞)

sβ σ p (ds).

Letting β ↑ p, we can replace β in this inequality by p. This shows that σ p ∈ M∞p (R). Next we claim that h(y)dy = (I p σ p )(dy), that is, h(y) = c p

(y,∞)

(s − y) p−1σ p (ds)

(7.9)

for a. e. y ∈ R.

If this is shown, then we obtain μ ∈ L p,α . The proof of (7.9) is as follows. Let τ (dy) = h(y)dy. Note that τ ∈ D(I 1 ). For large n we have I p−qn τ = I p−qn I qn σ qn = I p σ qn . As n → ∞, I p−qn τ tends to τ vaguely on R by Lemma 2.9. So, it is enough to show that I p σ qn → I p σ p

(vaguely on R),

n → ∞.

(7.10)

We write q for qn . Let f be a continuous function on R with support in [a, b]. We have

f (r)I p σ q (dr) = =

R R

f (r)dr

σ q (ds)

(r,∞) s −∞

c p (s − r) p−1σ q (ds)

c p (s − r) p−1 f (r)dr


=

=

R

σ q (ds)

(−∞,s0 ]

∞ 0

83

c p u p−1 f (s − u)du

σ q (ds) · · · +

(s0 ,∞)

σ q (ds) · · · = J1 + J2 .

If s0 is a continuity point of σ p , then J1 →

(−∞,s0 ]

σ p (ds)

∞

c p u p−1 f (s − u)du,

0

since σ q → σ p vaguely on R. Concerning J2 , we have ∞ s−a p−1 ≤ || f ||c p c u f (s − u)du u p−1du ∼ || f ||c p (b − a)s p−1, p 0 s−b

s → ∞.

Let 0 < ε 1. We have ∞ c

h(y)dy = cq+1 ≥ c p+1

(c,∞)

(s − c)q σ q (ds)

(2c,∞)

s p−ε

(s − c)q q σ (ds) ≥ c p+1 2−p s p−ε

(2c,∞)

s p−ε σ q (ds),

since, as s ↓ 2c, (s−c)q /s p−ε = (s−c)q−p+ε (1−c/s) p−ε decreases to 2−p+ε cq−p+ε ≥ 2−p+ε ≥ 2−p . It follows that ∞ c

h(y)dy ≥ c p+1 2−p

(2c,∞)

s p−1 σ q (ds),

which proves (7.11). Therefore J2 is uniformly small if s0 is close to ∞, and we obtain (7.10). Remark 7.10. Open question: Can one extend (7.8) to the case 0 < α < 2? As L0p,α and Lep,α are decreasing with respect to p, we define, for −∞ < α < 2, L0∞,α = L0∞,α (Rd ) =

L0p,α ,

(7.12)

Lep,α .

(7.13)

p>0

Le∞,α = Le∞,α (Rd ) =

p>0

These are described by L∞ and LE∞ , E ∈ B((0, 2)), introduced in Section 1.4.

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K. Sato

Theorem 7.11. Descriptions of Le∞,α for all α and L0∞,α for α = 1 are as follows. Le∞,α = L∞

for −∞ < α ≤ 0,

(7.14)

Le∞,α = L(∞α ,2)

for 0 < α < 2,

(7.15)

L0∞,α L0∞,α

for −∞ < α < 1,

= Le∞,α = {μ ∈

L(∞α ,2) :

Rd

xμ (dx) = 0}

(7.16) for 1 < α < 2.

(7.17)

Proof. (7.14) and (7.15): First, notice that Le∞,α = n=1,2,... Len,α . Let μ ∈ Le∞,α . Use Theorem 6.9. Then, for each n = 1, 2, . . ., νμ has a radial decomposition (n) (n) (λ (n) (d ξ ), u−α −1 hξ (log u)du), where hξ (y) is measurable in (ξ , y) and, for λ (n) (n)

a. e. ξ , hξ (y) is monotone of order n on R. It follows from Proposition 3.1 that we (n)

can choose λ (n) = λ and hξ = hξ independently of n. Thus, for λ -a. e. ξ , hξ (y) is completely monotone on R. We choose a modification of hξ (y) completely monotone on R for all ξ ∈ S. Further, we choose λ to be a probability measure. For y0 ∈ R, the function hξ (y0 + y), y ∈ R◦+ , is completely monotone on R◦+ and hence hξ (y0 + y) =

(0,∞)

e−yβ Γξy0 (d β ),

y>0

with a unique measure Γξy0 on (0, ∞) by Bernstein’s theorem (recall that our definiy tion of complete monotonicity involves hξ (y0 + y) → 0 as y → ∞, so that Γξ 0 has no mass at 0). In particular, we have Γξ0 for y0 = 0. If y0 < 0, then hξ (y) = hξ (y0 + (y − y0)) =

(0,∞)

e−(y−y0 )β Γξy0 (d β ),

y>0

and hence ey0 β Γξ 0 (d β ) = Γξ0 (d β ). Thus y

hξ (y0 + y) =

(0,∞)

Therefore hξ (y) =

e−(y0 +y)β Γξ0 (d β ),

(0,∞)

e−yβ Γξ0 (d β ),

y0 < 0, y > 0.

y ∈ R.

We see that {Γξ0 : ξ ∈ S} is a measurable family. Indeed, if Γξ0 is a continuous measure for every ξ , then it is proved from the inversion formula (see [55], p. 285) s 0

[ys]

(−y)m (d/dy)m (hξ (y)), m=0 m!

∑ y→∞

Γξ0 (d β ) = lim

s > 0,


85

where [ys] is the largest integer ≤ ys. If not, it is proved by approximating Γξ0 by the convolutions with continuous measures. We have ∞>

|x|≤1

|x|2 νμ (dx) = = = =

S

S

S

S

λ (d ξ ) λ (d ξ ) λ (d ξ ) λ (d ξ )

1 0

1

u1−α hξ (log u)du u1−α du

0

1

(0,∞)

udu

(α ,∞)

0

(α ,∞)

Γξ (d β )

u−β Γξ0 (d β )

u−β Γξ (d β ) 1

u1−β du,

0

where we define

Γξ (E) = Since

(0,∞)

1E (α + β )Γξ0 (d β ),

E ∈ B((α , ∞)).

1 1−β du = ∞ for β ≥ 2, we obtain Γξ ([2, ∞)) = 0 for λ -a. e. ξ . We have 0 u |x|≤1

|x|2 νμ (dx) =

S

λ (d ξ )

(α ,2)

(2 − β )−1Γξ (d β ).

We also have ∞>

|x|>1

νμ (dx) = = = =

S

S

S

S

λ (d ξ ) λ (d ξ ) λ (d ξ ) λ (d ξ )

∞ 1

∞

u−α −1 hξ (log u)du u−α −1 du

1

∞

u−1 du

1

(α ,2)

(0,∞)

u−β Γξ (d β )

(α ,2) ∞

Γξ (d β )

u−β Γξ0 (d β )

u−β −1du,

1

and 1∞ u−β −1du = ∞ for β ≤ 0. Hence, if α < 0, then Γξ ((α , 0]) = 0 for λ -a. e. ξ . For any α < 2 we have |x|>1

νμ (dx) =

S

λ (d ξ )

(α ∨0,2)

β −1Γξ (d β ).

Similarly, it follows from

νμ (B) =

S

λ (d ξ )

∞ 0

1B (uξ )u−α −1hξ (log u)du

(7.18)

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K. Sato

that

νμ (B) =

S

λ (d ξ )

(α ∨0,2)

Γξ (d β )

∞ 0

1B (uξ )u−β −1du.

(7.19)

The measure λ (d ξ )Γξ (d β ) on S × (α ∨ 0, 2) is written to Γ (d β )λβ (d ξ ), where Γ (d β ) is a measure on (α ∨ 0, 2) satisfying (α ∨0,2)

(β −1 + (2 − β )−1)Γ (d β ) < ∞

and {λβ : β ∈ (α ∨ 0, 2)} is a measurable family of probability measures on S. (α ∨0,2)

Therefore Le∞,α ⊂ L∞

. (α ∨0,2)

Conversely, suppose that μ ∈ L∞ with Lévy measure νμ satisfying (1.6). Then, defining λ (d ξ ) and Γξ (d β ) in the converse direction and letting hξ (y) = −y(β −α )Γ (d β ), we see (7.19) and then (7.18) with h (y) completely ξ ξ (α ∨0,2) e monotone on R. Hence μ ∈ Le∞,α . This completes the proof of (7.14) and (7.15). Assertions (7.16) and (7.17) follow from Theorem 6.12 (i) and (ii), respectively. (α ,2) Note that if μ ∈ L∞ with 1 < α < 2, then |x|>1

|x|νμ (dx) =

and hence

Rd

(α ,2)

Γ (d β )

S

λβ (d ξ )

∞ 1

r−β dr =

(α ,2)

(β − 1)−1Γ (d β ) < ∞

|x|μ (dx) < ∞.

Remark 7.12. Open problem: Give the description of the class L0∞,1 . Remark 7.13. Open question: Does there exist a function f (s), s ≥ 0, such that L0∞,α or Le∞,α is equal to R(Φ f ), R0 (Φ f ), or Re (Φ f )? In particular, for L∞ = L0∞,0 = Le∞,0 , this is a long-standing question. Theorem 7.14. We have e e K∞, α L∞,α 0 K∞, α

L0∞,α

for −∞ < α < 2

(7.20)

for α ∈ (−∞, 1) ∪ (1, 2).

(7.21)

e −α −1 Proof. We know that μ ∈ K∞, α if and only if νμ has radial decomposition (λ , u ◦ e kξ (u)du) with kξ (u) completely monotone on R+ . On the other hand, μ ∈ L∞,α if and only if νμ has radial decomposition (λ , u−α −1 hξ (log u)du) with hξ (y) completely monotone on R. Since the complete monotonicity of hξ (y) on R implies that e e of hξ (log u) on R◦+ , we have K∞, α ⊃ L∞,α . To see the strictness of the inclusion, use y −ce and k(u) = h(log u) = e−cu with c > 0; k(u) is completely the functions h(y) = e ◦ monotone on R+ but h(y) is not completely monotone on R, since

h (y) = −h(y)cey (1 − cey ) < 0 for y close to −∞. Hence (7.20) is true.


87

Assertion (7.21) for α ∈ (−∞, 1) is automatic from (7.20). For α ∈ (1, 2), combine (7.20) with the condition of zero mean. When μ ∈ L∞ , let Γμ denote the measure Γ in the representation (1.6) of νμ . We give some moment properties of distributions in L∞ . Proposition 7.15. Let μ ∈ L∞ . Let 0 < α < 2.

(i) If Γμ ((0, α ]) > 0, then Rd |x|α μ (dx) = ∞. (ii) Suppose that Γμ ((0, α ]) = 0. Then, Rd |x|α μ (dx) < ∞ if and only if (α ,2) (β − α )−1Γμ (d β ) < ∞. Proof. Since λβ in (1.6) satisfies λβ (S) = 1, we have |x|>1

|x|α νμ (dx) =

(0,2)

Γμ (d β )

∞

rα −β −1 dr.

1

Since 1∞ rα −β −1 dr is infinite for β ≤ α and (β − α )−1 for β > α , our assertions follow. Proposition 7.16. (i) Let μ ∈ L∞ and suppose that μ is not Gaussian (that is, ν = 0). Let α0 be the infimum of the support of Γμ . Then α0 ∈ [0, 2) and μ α α Rd |x| μ (dx) = ∞ for α ∈ (α0 , 2). If α0 > 0, then Rd |x| μ (dx) < ∞ for α ∈ (0, α0 ). (α ,2) (ii) Let 0 < α < 2. There exists μ ∈ L∞ such that Rd |x|α μ (dx) = ∞. Proof. Assertion (i) follows from Proposition 7.15. To see (ii), choose α ∈ (α , 2), let Γ (d β ) = 1(α ,α ) (β )d β , and use Proposition 7.15 (ii). Remark 7.17. The identity (7.5) expresses the iteration of Λ p,α for α = 1. The iteration of a stochastic integral mapping Φ f generates nested classes of their ranges. The description of their intersection is an interesting problem. See Maejima and Sato [27] and the references therein.

7.2 Lp,α , L0p,α , and Lep,α for α ∈ (−∞, 2) with Fixed p Little is known about the one-parameter families {L p,α : α ∈ (−∞, 2)}, {L0p,α : α ∈ (−∞, 2)}, and {Lep,α : α ∈ (−∞, 2)} for fixed p. Lemma 7.18. Let n be a positive integer. If f (r) is monotone of order n on R, then, for any a > 0, e−ar f (r) is monotone of order n on R. Proof. This follows from Lemma 5.17, as e−ar is completely monotone on R.

Theorem 7.19. Let n be a positive integer. Then, for −∞ < α < α < 2, Ln,α Ln,α ,

L0n,α L0n,α ,

and Len,α Len,α .

(7.22)

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K. Sato

Proof. Step 1. Let us prove that Len,α ⊃ Len,α . Let μ ∈ Len,α . Then νμ has radial

decomposition (λ (d ξ ), u−α −1 hξ (log u)du) with hξ (y) monotone of order n on R. Let

hξ (y) = e−(α −α )y hξ (y),

Then hξ (y) is monotone of order n on R by the lemma above, and u−α −1 hξ (log u) =

u−α −1 hξ (log u). Hence μ ∈ Len,α . Step 2. Let us prove Ln,α ⊃ Ln,α and L0n,α ⊃ L0n,α . If α < 1, then these follow from Step 1. Suppose α = 1 and let μ ∈ Ln,α = L0n,α . Then, Rd |x|μ (dx) < ∞ and e L ). 1, νμ ∈ R(Λn,1 Rd x μ (dx) = 0 from Theorem 6.12 (ii). Since μ ∈ Ln,1 from Step 0 L L 0 0 Thus there is ν ∈ D(Λn,1 ) such that νμ = Λn,1 ν . We have |x|>1 |x|ν (dx) < ∞ from Theorem 6.2. Since |x|>1 |x|νμ (dx) < ∞, we have ∞

ds 0

|ln,1 (s)x|>1

|ln,1 (s)x|ν 0 (dx) < ∞.

Moreover,

γμ = −

|x|>1

xνμ (dx) = −

∞

ds 0

|ln,1 (s)x|>1

ln,1 (s)xν 0 (dx).

−1 Aμ , and γρ = − |x|>1 x Choose ρ ∈ ID such that νρ = ν 0 , Aρ = 0∞ ln,1 (s)2 ds ν 0 (dx). Then it follows from Proposition 3.18 that ρ ∈ D0 (Λn,1 ) and Λn,1 ρ = μ . Hence μ ∈ L0n,1 ⊂ Ln,1 . Similarly, if α > 1 and if μ ∈ Ln,α = L0n,α , then μ ∈ Ln,α = L0n,α . Step 3. To show the strictness of the inclusion, let λ be a non-zero finite measure on S and let h(y) = (−y)n−1 1(−∞,0) (y), which is monotone of order n on R (Example 2.17 (a)). Then (λ (d ξ ), u−α −1 h(log u)du) is a radial decomposition of a Lévy mea sure ν , since 01 u1−α h(log u)du < ∞. Let μ ∈ ID with νμ = ν . Then μ ∈ Len,α but μ ∈ Len,α , as is seen by an argument similar to the proof of Theorem 5.19. Indeed, we have u−α −1h(log u) = u−α −1 h (log u) for

h (y) = e(α −α )y (−y)n−1 1(−∞,0)(y), which is not monotone of any order on R from Proposition 2.13 (iii). Strictness of the first and second inclusions in (7.22) is obtained from that of the third. Remark 7.20. Open question: Is (7.22) true for p ∈ R◦+ in place of n? Acknowledgments The author thanks Makoto Maejima and V´ıctor Pérez-Abreu for their constant encouragement by proposing and exploring many problems related to the subject and for their valuable comments during preparation of this work, and Yohei Ueda for his helpful remarks for improvement of Section 6.1.


89

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Packing and Hausdorff Measures of Stable Trees Thomas Duquesne

Abstract In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of Lévy trees (introduced by Le Gall and Le Jan in [33]) that contains Aldous’s continuum random tree which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for level sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from [14]. AMS Subject Classification 2000: Primary: 60G57, 60J80 Secondary: 28A78 Keywords Hausdorff measure · Lévy trees · Local time measure · Mass measure · Packing measure · Stable trees

1 Introduction Stable trees are particular instances of Lévy trees that form a class of random compact metric spaces introduced by Le Gall and Le Jan in [33] as the genealogy of Continuous State Branching Processes (CSBP for short). The class of stable trees contains Aldous’ continuum random tree that corresponds to the Brownian case (see [2, 3]). Stable trees (and more generally Lévy trees) are the scaling limit of GaltonWatson trees (see [12] Chapter 2 and [10]). Various geometric and distributional T. Duquesne () Université Paris 6 – Pierre et Marie Curie, Laboratoire de Probabilités et Modèles Aléatoires, Case courrier 188, 4 Place Jussieu, 75252 Paris CX 05, France e-mail: [email protected]


93

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properties of Lévy trees (and of stable trees, consequently) have been studied in [13] and in Weill [40]. An alternative construction of Lévy trees is discussed in [16]. Stable trees have been also studied in connection with fragmentation processes: see Miermont [35, 36], Haas and Miermont [23], Goldschmidt and Haas [21] for the stable cases and see Abraham and Delmas [1] for related models concerning more general Lévy trees. Fractal properties of stable trees have been discussed in [13] and [14]: Hausdorff and packing dimensions of stable trees are computed in [13] and the exact Hausdorff measure of Aldous’ continuum random tree is given in [14]. The same paper contains partial results for the non-Brownian stable trees that suggest there is no exact Hausdorff measure in these cases. In this paper we prove there is no exact packing measure for the level sets of stable trees (including the Brownian case) and we also prove that there is no exact Hausdorff measure with regularly varying gauge function for the non-Brownian stable trees and their level sets. Before stating the main results of the paper, let us recall the definition of stable CSBPs and the definition of stable trees that represent the genealogy of stable CSBPs. CSBPs are time- and space-continuous analogues of Galton-Watson Markov chains. They have been introduced by Jirina [26] and Lamperti [30] as the [0, ∞]-valued Feller processes that are absorbed in states {0} and {∞} and whose kernel semi-group (pt (x, dy); x ∈ [0, ∞], t ∈ [0, ∞)) enjoys the branching property: pt (x, ·) ∗ pt (x , ·) = pt (x + x , ·), for every x, x ∈ [0, ∞] and every t ∈ [0, ∞). As pointed out in Lamperti [30], CSBPs are time-changed spectrally positive Lévy processes. Namely, let Y = (Yt ,t ≥ 0) be a Lévy process starting at 0 that is defined on a probability space (Ω , F , P) and that has no positive jump. Let x ∈ (0, ∞). Set At = inf{s ≥ 0 : 0s du/(Yu + x) > t} for any t ≥ 0, and Tx = inf{s ≥ 0 : Ys = −x}, with the convention that inf 0/ = ∞. Next set Zt = XAt ∧Tx if At ∧ Tx is finite and set Zt = ∞ if not. Then, Z = (Zt ,t ≥ 0) is a CSBP with initial state x (see Helland [25] for a proof in the conservative cases). Recall that the distribution of Y is characterized by its Laplace exponent ψ given by E[exp(−λ Yt )] = exp(t ψ (λ )), t, λ ≥ 0 (see Bertoin [5], Chapter 7). Consequently, the law of the CSBP Z is also characterised by ψ and it is called its branching mechanism. We shall restrict to γ -stable CSBPs for which ψ (λ ) = λ γ , λ ≥ 0, where γ ∈ (1, 2]. The case γ = 2 shall be referred to as the Brownian case (and the corresponding CSBP is the Feller diffusion) and the cases 1 < γ < 2 shall be referred to as the non-Brownian stable cases. Let Z be a γ -stable CSBP defined on (Ω , F , P). As a consequence of a result due to Silverstein [38], the kernel semigroup of Z is characterised as follows: for any λ , s,t ≥ 0, one has E[exp(−λ Zt+s )|Zs ] = exp(−Zs u(t, λ )), where u(t, λ ) is the unique nonnegative solution of ∂ u(t, λ )/∂ t = −u(t, λ )γ and u(0, λ ) = λ . This ordinary differential equation can be explicitly solved as follows. u(t, λ ) = (γ −1)t +

1

λ γ −1

−

1 γ −1

,

t, λ ≥ 0.

(1.1)

Packing and Hausdorff Measures of Stable Trees

95

It is easy to deduce from this formula that γ -stable CSBPs get almost surely extinct in finite time with probability one: P(∃t ≥ 0 : Zt = 0) = 1. We refer to Bingham [6] for more details on CSBPs. Lévy trees have been introduced by Le Gall and Le Jan in [33] via a coding function called the height process whose definition is recalled in Section 2.2. Let us briefly recall the formalism discussed in [13] where Lévy trees are viewed as random variables taking values in the space of all compact rooted R-trees. Informally, an R-tree is a metric space (T , d) such that for any two points σ and σ in T there is a unique arc with endpoints σ and σ and this arc is isometric to a compact interval of the real line. A rooted R-tree is a R-tree with a distinguished point that we denote by ρ and that we call the root. We say that two rooted R-trees are equivalent if there is a root-preserving isometry that maps one onto the other. Instead of considering all compact rooted R-trees, we introduce the set T of equivalence classes of compact rooted R-trees. Evans, Pitman and Winter in [19] noticed that T equipped with the Gromov-Hausdorff distance [22], is a Polish space (see Section 2.2 for more details). With any stable exponent γ ∈ (1, 2] one can associate a sigma-finite measure Θγ on T called the “law” of the γ -stable tree. Although Θγ is an infinite measure, one can prove the following: Define Γ (T ) = supσ ∈T d(ρ , σ ) that is the total height of T . Then, for any a ∈ (0, ∞), one has 1

Θγ (Γ (T ) > a) = ((γ −1)a)− γ −1 . Stable trees enjoy the so-called branching property, that obviously holds true for Galton-Watson trees. More precisely, for every a > 0, under the probability measure Θγ ( · | Γ (T ) > a) and conditionally given the part of T below level a, the subtrees above level a are distributed as the atoms of a Poisson point measure whose intensity is a random multiple of Θγ , and the random factor is the total mass of the a-local time measure that is defined below (see Section 2.2 for a precise definition). It is important to mention that Weill in [40] proves that the branching property characterizes Lévy trees, and therefore stable trees. We now define Θγ by an approximation with Galton-Watson trees as follows. Let ξ be a probability distribution on the set of nonnegative integers N. We first assume that ∑k≥0 kξ (k) = 1 and that ξ is in the domain of attraction of a γ -stable distribution. More precisely, let Y1 be a random variable such that log E[exp(−λ Y1 )] = λ γ , for any λ ∈ [0, ∞). Let (Jk , k ≥ 0) be an i.i.d. sequence of r.v. with law ξ . We assume there exists an increasing sequence (a p , p ≥ 0) of positive integers such that (a p )−1 (J1 + · · · + J p − p) converges in distribution to Y1 . Denote by τ a GaltonWatson tree with offspring distribution ξ that can be viewed as a random rooted R-tree (τ , δ , ρ ) by affecting length 1 to each edge. Thus, (τ , 1p δ , ρ ) is the tree τ whose edges are rescaled by a factor 1/p and we simply denote it by 1p τ . Then, for any a ∈ (0, ∞), the law of 1p τ under P( · | 1p Γ (τ ) > a) converges weakly in T to the probability distribution Θγ ( · | Γ (T ) > a), when p goes to ∞. This result is Theorem 4.1 [13]. Let us introduce two important kinds of measures defined on γ -stable trees. Let (T , d, ρ ) be a γ -stable tree. For every a > 0, we define the a-level set T (a) of T

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as the set of points that are at distance a from the root. Namely, T (a) := σ ∈ T : d(ρ , σ ) = a .

(1.2)

We then define the random measure a on T (a) in the following way. For every ε > 0, write Tε (a) for the finite subset of T (a) consisting of those vertices that have descendants at level a + ε . Then, Θγ -a.e. for every bounded continuous function f on T , we have 1 (1.3) a , f = lim ((γ −1)ε ) γ −1 ∑ f (σ ). ε ↓0

σ ∈Tε (a)

The measure a is a finite measure on T (a) that is called the a-local time measure of T . We refer to [13] Section 4.2 for the construction and the main properties of the local time measures (a , a ≥ 0) (see also Section 2.2 for more details). Theorem 4.3 [13] ensures we can choose a modification of the local time measures (a , a ≥ 0) in such a way that a → a is Θγ -a.e. cadlag for the weak topology on the space of finite measures on T . We next define the mass measure m on the tree T by m=

∞ 0

da a .

(1.4)

The topological support of m is T . Note that the definitions of the local time measures and of the mass measure only involve the metric properties of T . Let us mention that γ -stable trees enjoy the following scaling property: For any c ∈ (0, ∞), the “law” of (T , c d, ρ ) under Θγ is c1/(γ −1) Θγ . Then, it is easy to show that for any a, c ∈ (0, ∞) the law of c1/(γ −1) a/c under Θγ is the law of a under c1/(γ −1) Θγ (here, b stands for the total mass of the b-local time measure). Similarly, the law of cγ /(γ −1) m under Θγ is the law of m under c1/(γ −1) Θγ . Since a and m are in some sense the most spread out measures on respectively T (a) and T , these scaling properties give a heuristic explanation for the following results that concern the fractal dimensions of stable trees (see [13] for a proof): For / the Hausdorff and the packing dimensions of any a ∈ (0, ∞), Θγ -a.e. on {T (a) = 0} T (a) are equal to 1/(γ − 1) and Θγ -a.e. the Hausdorff and the packing dimensions of T are equal to γ /(γ − 1). In this paper we discuss finer results concerning possible exact Hausdorff and packing measures for stable trees and their level sets. We first state a result concerning the exact packing measure for level sets. To that end, let us briefly recall the definition of packing measures. Packing measures have been introduced by Taylor and Tricot in [39]. Though their construction is done in Euclidian spaces, it easily extends to metric spaces and more specifically to γ -stable trees. More precisely, for ¯ σ , r) (resp. B(σ , r)) the closed any σ ∈ T and any r ∈ [0, ∞), let us denote by B( (resp. open) ball of T with center σ and radius r. Let A ⊂ T and ε ∈ (0, ∞). An ¯ n , rn ), ε -packing of A is a countable collection of pairwise disjoint closed balls B(x n ≥ 0, such that xn ∈ A and rn ≤ ε . We restrict our attention to packing measures associated with a regular gauge function in the following sense: A function


97

g : (0, r0 ) → (0, ∞) is a regular gauge function if it is continuous, non decreasing, if lim0+ g = 0 and if there exists a constant C ∈ (1, ∞) such that ∃C > 1 :

g(2r) ≤ Cg(r) ,

r ∈ (0, r0 /2).

(1.5)

Such a property shall be referred to as a C-doubling condition. We then set Pg∗ (A) = lim sup ε ↓0

¯ n , rn ), n ≥ 0) ε − packing of A ∑ g(rn ); (B(x

(1.6)

n≥0

that is the g-packing pre-measure of A and we define the g-packing outer measure of A as

(1.7) Pg (A) = inf ∑ Pg∗ (En ); A ⊂ En . n≥0

n≥0

As in Euclidian spaces, Pg is a Borel regular metric outer measure (see Section 2.1 for more details). The original definition of packing measures [39] makes use, as set function, of the diameter of open ball packing instead of the radius of closed ball packing. As pointed out by H. Haase [24], diameter-type packing measures may be not Borel regular: H. Joyce [28] provides an explicit example where this problem occurs. In our setting, we do not face such a problem and our results hold true for diameter-type and radius-type packing measures as well thanks to specific properties of compact real trees (see Section 2.1 for a brief discussion). The following theorem shows that the level sets of stable trees have no exact packing measure, even in the Brownian case. Theorem 1.1. Let γ ∈ (1, 2] and let us consider a γ -stable tree (T , d, ρ ) under its excursion measure Θγ . Let g : (0, 1) → (0, ∞) be any function such that lim r

r→0

1 − γ −1

g(r) = 0.

(1.8)

n γ (i) If ∑n≥1 2 γ −1 g(2−n) < ∞, then for any a ∈ (0, ∞), Θγ -a.e. on {T (a) = 0} / a and for -almost all σ , we have lim inf n→∞

a (B(σ , 2−n )) = ∞. g(2−n )

(1.9)

Moreover, if g is a regular gauge function, then Pg (T (a) ) = 0, Θγ -a.e. n γ (ii) If ∑n≥1 2 γ −1 g(2−n ) = ∞, then for any a ∈ (0, ∞), Θγ -a.e. and for a -almost all σ , we have a (B(σ , 2−n )) lim inf =0. (1.10) n→∞ g(2−n ) Moreover, if g is a regular gauge function, then Pg (T (a) ) = ∞, Θγ -a.e. on the event {T (a) = 0}. /

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This result is not surprising, even in the Brownian case, for it has been proved in [34] that super-Brownian motion with quadratic branching mechanism has no exact packing measure in the super-critical dimension d ≥ 3 and [34] provides a test that is close in some sense to the test given in the previous theorem. Remark 1.2. For any p ≥ 1, define recursively the functions log p by log1 = log and log p+1 = log p ◦ log. The previous theorem provides the following family of critical gauge functions for packing measures of level sets of a γ -stable tree: For any θ ∈ R and any p ≥ 1, set 1

g p,θ (r) =

r γ −1 1

(log(1/r) . . . log p (1/r)) γ (log p+1 (1/r))θ

.

If γθ > 1, then for any a ∈ (0, ∞), one has Pg p,θ (T (a) ) = 0, Θγ -a.e. and if γθ ≤ 1, / then for any a ∈ (0, ∞), one has Pg p,θ (T (a) ) = ∞, Θγ -a.e. on the event {T (a) = 0}. Remark 1.3. Although the level sets of stable trees have no exact packing measure, the whole γ -stable tree has an exact packing measure as shown in the preprint [11]. More precisely, for any r ∈ (0, 1/e), we set γ

g(r) =

r γ −1 1

(loglog 1/r) γ −1

.

Then, there exists c0 ∈ (0, ∞) such that Pg = c0 m, Θγ -a.e. Remark 1.4. Note that no other condition than (1.8) is imposed on g for the test on the lower density of a to hold true. A similar test holds true when the sequence of radii (2−n , n ≥ 0) is replaced by (sR−n , n ≥ 0), with s ∈ ]0, 1[, and R > 1. As pointed out by Berlinkov (Lemma 1 [4]), the result can rephrased under the form of an integral test: indeed, if we assume that g : (0, 1) → (0, ∞) is non-decreasing and that it satisfies (1.8) and the doubling condition (1.5), then, Theorem 1.1 easily implies the following integral test.

γ

(i) If 0+ g(r) / and for r dr < ∞, then for any a ∈ (0, ∞), Θγ -a.e. on {T (a) = 0} a -almost all σ , we have lim inf r↓0

(ii) If

0+

g(r)γ r

a (B(σ , r)) = ∞. g(r)

dr = ∞, then for any a ∈ (0, ∞), Θγ -a.e. for a -almost all σ , we have lim inf r↓0

a (B(σ , r)) = 0. g(r)

Let us briefly recall the definition of Hausdorff measures on a γ -stable tree (T , d). Let us fix a regular gauge function g. For any subset E ⊂ T , we set


99

diam(E) = supx,y∈E d(x, y) that is the diameter of E. For any A ⊂ T , the g-Hausdorff measure of A is then given by Hg (A) = lim inf ε ↓0

∑ g(diam(En )); diam(En ) < ε and A ⊂

n≥0

En .

(1.11)

n≥0

As in the Euclidian case, Hg is a metric and Borel regular outer measure on T . In the Brownian case Theorem 1.3 in [14] asserts that there exists a constant c1 ∈ (0, ∞) such that for any a ∈ (0, ∞), Θ2 -a.e. we have Hg1 ( · ∩ T (a) ) = c1 a , where g1 (r) = r log log 1/r. The non-Brownian stable cases are quite different as shown by the following proposition that asserts that in these cases, there is no exact upperdensity for local time measures. Let us mention that the first point of the theorem is proved in Proposition 5.2 [14]. Proposition 1.5. Let γ ∈ (1, 2) and let (T , d, ρ ) be a γ -stable tree under its excursion measure Θγ . Let g : (0, 1) → (0, ∞) be a function such that lim g(r) = 0

and

r→0

lim r

r→0

1 − γ −1

g(r) = ∞.

(1.12)

−n

(i) (Prop. 5.2 [14]) If ∑n≥1 g(22−n )γ −1 < ∞, then for any a ∈ (0, ∞), Θγ -a.e. for a -almost all σ , we have lim sup n→∞

a (B(σ , 2−n )) = 0. g(2−n)

(1.13)

Moreover, if g is a regular gauge function, then Hg (T (a) ) = ∞, Θγ -a.e. on the event {T (a) = 0}. / −n / (ii) If ∑n≥1 g(22−n )γ −1 = ∞, then for any a ∈ (0, ∞), Θγ -a.e. on the event {T (a) = 0} a and for -almost all σ , we have lim sup n→∞

a (B(σ , 2−n )) = ∞. g(2−n )

(1.14)

Recall that a function g is regularly varying at 0 with exponent q iff for any c ∈ (0, ∞), g(cr)/g(r) tend to cq when r goes to 0. Theorem 1.6. Let γ ∈ (1, 2) and let (T , d, ρ ) be a γ -stable tree under its excursion measure Θγ . Then the level sets of T have no exact Hausdorff measure with continuous regularly varying gauge function. More precisely, let g : (0, 1) → (0, ∞) be a regular gauge function that is regularly varying at 0. / – Either for any a ∈ (0, ∞), we Θγ -a.e. have Hg (T (a)) = ∞, on {T (a) = 0}, – or for any a ∈ (0, ∞), we Θγ -a.e. have Hg (T (a)) = 0.

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Remark 1.7. Proposition 1.5 and Theorem 1.6 suggest that if 2−n

∑ g(2−n)γ−1

= ∞,

n≥1

then, Hg (T (a)) = 0, Θγ -a.e. as conjectured in [14]. The best result in this direction is Theorem 1.5 in [14] that shows that Hg (T (a)) = 0, Θγ -a.e. if g is of the following form: g(r) = r

1 − γ −1

1

1

1

(log r ) γ −1 (loglog r )u ,

with u < 0. Remark 1.8. As in Remark 1.4, note that no other condition than (1.12) is imposed on g for the test on the upper density of a to hold true. A similar test holds true when the sequence of radii (2−n , n ≥ 0) is replaced by (sR−n , n ≥ 0), with s ∈ ]0, 1[, and R > 1. If we assume that g : (0, 1) → (0, ∞) is non-decreasing and that it satisfies (1.12) and the doubling condition (1.5), then, Proposition 1.5 easily implies the following integral test. (i) If

dr 0+ g(r)γ −1

< ∞, then for any a ∈ (0, ∞), Θγ -a.e. for a -almost all σ , we have lim sup r↓0

a (B(σ , r)) = 0. g(r)

(ii) If 0+ g(r)drγ −1 = ∞, then for any a ∈ (0, ∞), Θγ -a.e. on {T (a) = 0} / and for a -almost all σ , we have lim sup r↓0

a (B(σ , r)) = ∞. g(r)

Let us discuss now the Hausdorff properties of whole stable trees. In the Brownian case, Theorem 1.1 in [14] asserts that there exists a constant c2 ∈ (0, ∞) such that Θ2 -a.e. we have Hg2 = c2 m, where g2 (r) = r2 log log 1/r. In the nonBrownian stable cases, the situation is quite different as shown by the following proposition that asserts that in these cases, the mass measure has no exact upper-density. Let us mention that the first point of the theorem is proved in Proposition 5.1 [14]. Proposition 1.9. Let γ ∈ (1, 2) and let (T , d, ρ ) be a γ -stable tree under its excursion measure Θγ . Let g : (0, 1) → (0, ∞) be a function such that lim g(r) = 0

r→0

and

lim r

r→0

γ − γ −1

g(r) = ∞.

(1.15)

−γ n

(i) (Prop. 5.1 [14]) If ∑n≥1 g(22−n )γ −1 < ∞, then Θγ -a.e. for m-almost all σ , we have lim sup n→∞

m(B(σ , 2−n )) = 0. g(2−n)

(1.16)


101

Moreover, if g is a regular gauge function, then Hg (T ) = ∞, Θγ -a.e. −γ n (ii) If ∑n≥1 g(22−n )γ −1 = ∞, then Θγ -a.e. for m-almost all σ , we have lim sup n→∞

m(B(σ , 2−n )) = ∞. g(2−n )

(1.17)

The previous proposition is completed by the following result. Theorem 1.10. Let γ ∈ (1, 2) and let (T , d, ρ ) be a γ -stable tree under its excursion measure Θγ . Then T has no exact Hausdorff measure with continuous regularly varying gauge function. More precisely, let g : (0, 1) → (0, ∞) be a regular gauge function that is regularly varying at 0. – Either Hg (T ) = ∞, Θγ -a.e. – or Hg (T ) = 0, Θγ -a.e. Remark 1.11. Proposition 1.9 and Theorem 1.10 suggest that if 2 −γ n ∑ −n γ−1 = ∞ n≥1 g(2 ) then, Hg (T ) = 0, Θγ -a.e. as conjectured in [14]. The best result in this direction is Theorem 1.4 in [14] that show that Hg (T ) = 0, Θγ -a.e. if g is of the following form: g(r) = r

γ

− γ −1

1

1

1

(log r ) γ −1 (loglog r )u ,

with u < 0. Remark 1.12. As in Remarks 1.4 and 1.8, note that no other condition than (1.15) is imposed on g for the test on the upper density of a to hold true. A similar test holds true when the sequence of radii (2−n , n ≥ 0) is replaced by (sR−n , n ≥ 0), with s ∈ ]0, 1[, and R > 1. If we assume that g : (0, 1) → (0, ∞) is non-decreasing and that it satisfies (1.15) and the doubling condition (1.5), then, Proposition 1.9 easily implies the following integral test. (i) If

γ −1 dr 0+ (r/g(r))

< ∞, then, Θγ -a.e. for m-almost all σ , we have lim sup r↓0

(ii) If

γ −1 dr 0+ (r/g(r))

m(B(σ , r)) = 0. g(r)

= ∞, then, Θγ -a.e. for m-almost all σ , we have lim sup r↓0

m(B(σ , r)) = ∞. g(r)

The paper is organised as follows. In Section 2.1, we recall the basic comparison results on Hausdorff and packing measures in metric spaces. In Section 2.2, we

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introduce the γ -stable height processes and the γ -stable trees, and we recall a key decomposition of stable trees according the ancestral line of a randomly chosen vertex that is used to prove the upper- and lower-density results for the local time measures and the mass measure. In Section 2.3, we state various estimates that are used in the proof sections. Section 3 is devoted to the proofs of the main results of the paper.

2 Notation, Definitions and Preliminary Results 2.1 Hausdorff and Packing Measures on Metric Spaces Though standard in Euclidian spaces (see Taylor and Tricot [39]), packing measures are less usual in Polish spaces that is why we briefly recall few results in this section. As already mentioned, we restrict our attention to continuous gauge functions that satisfy a doubling condition: Let us fix C > 1. We denote by GC the set of such

regular gauge functions that satisfy a C-doubling condition and we set G = C>1 GC that is the set of the gauge functions we shall consider. Let us mention that, instead of regular gauge functions, some authors speak of blanketed Hausdorff functions after Larman [31]. Let (T , d) be an uncountable complete and separable metric space. Let us fix g ∈ GC . Recall from (1.7) the definition of the g-packing measure Pg and from (1.11) the definition of the g-Hausdorff measure Hg . We shall use the following comparison results. Lemma 2.1. (Taylor and Tricot [39], Edgar [17]). Let g ∈ GC . Then, for any finite Borel measure μ on T and for any Borel subset A of T , the following holds true. μ (B(σ ,r)) ≤ 1, for any σ ∈ A, then Pg (A) ≥ C−2 μ (A). g(r) σ ,r)) lim infr→0 μ (B( ≥ 1, for any σ ∈ A, then Pg (A) ≤ μ (A). g(r) μ (B(σ ,r)) lim supr→0 g(r) ≤ 1, for any σ ∈ A, then Hg (A) ≥ C−1 μ (A). σ ,r)) lim supr→0 μ (B( ≥ 1, for any σ ∈ A, then Hg (A) ≤ C μ (A). g(r)

(i) If lim infr→0 (ii) If (iii) If (iv) If

Points (iii) and (iv) in Euclidian spaces are stated in Lemmas 2 and 3 in Rogers and Taylor [37]. Points (i) and (ii) in Euclidian spaces can be found in Theorem 5.4 in Taylor and Tricot [39]. We refer to Edgar [17] for a proof of Lemma 2.1 for general metric spaces: For (i) and (ii), see Theorem 4.15 [17] in combination with Proposition 4.24 [17]. For (iii) and (iv), see Theorem 5.9 [17]. Remark 2.2. As already mentioned, the original definition of packing measures [39] uses the diameter of open ball packing instead of the radius of closed ball packing. As pointed out by H. Haase [24], diameter-type packing measures may be not Borel regular: H. Joyce [28] provides an explicit example where this problem occurs (see also H. Joyce [27] and H. Joyce and D. Preiss [29]). Let us briefly explain why there


103

is no such problem when dealing with stable trees and more generally with compact real trees: we now assume that (T , d) is a compact real tree with a distinguished point ρ that is called the root. In real trees, any two points σ and σ are joined by a unique arc with endpoints σ and σ and this arc is isometric to a compact interval of the real line (see Definition 2.3). Let A ⊂ T and ε ∈ (0, ∞). Let us say that an ε -open ball packing of A is a countable collection of pairwise disjoint open balls B(σn , rn ), n ≥ 0, such that σn ∈ A and diam(B(σn , rn )) ≤ ε . Let g be a regular gauge function that satisfies a C-doubling condition. For any ε , we set g(ε ) (A) = sup P

∑g

diam(B(σn , rn )) ; (B(σn , rn ))n≥0 , ε -op. ball pack. of A

n≥0

g(ε ) (A) and P g (A) = inf ∗ (A) = lim P P g ε ↓0

∗ (En ); A ⊂ ∑P g

n≥0

En .

n≥0

g is a Following exactly the proof of Edgar Proposition 5.2 [17], one sees that P metric outer measure. Let us now briefly explain why it is equivalent to the radiustype measure using closed ball packing, as defined previously by (1.6) and (1.7). Let us recall notation Γ (T ) = sup{d(ρ , σ ) ; σ ∈ T } that is the total height of T . Next, let σ ∈ T and r ∈ (0, ∞). We denote the actual radius of the open ball B(σ , r) by R(B(σ , r)) = sup{d(σ , σ ) ; σ ∈ B(σ , r)}. Note that R(B(σ , r)) = r

=⇒

r ≤ diam(B(σ , r)) ≤ 2r.

(2.1)

If there exists σ0 ∈ T such that d(σ0 , σ ) ≥ r, then R(B(σ , r)) = r, since we can find points on the unique geodesic joining σ to σ0 that are at distance r from σ , for any r < r. If R(B(σ , r)) < r, we therefore see that Γ (T ) < 2r. By (2.1), we then get 1

∀σ ∈ T , ∀ 0 < r < 2 Γ (T ) ,

r ≤ diam(B(σ , r)) ≤ 2r .

Using the continuity of g to control the difference between closed and open ball packings, we finally get that ∀A ⊂ T ,

g (A) ≤ C Pg (A). Pg (A) ≤ P

g and it is clear that Theorem 1.1 holds Since Pg is a Borel regular measure, so is P true for diameter-type packing measures.

2.2 Height Processes and Lévy Trees In this section we recall (mostly from [12] and [13]) various results concerning stable height processes and stable trees that are used in Sections 2.3 and in the proof sections.

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The Height Process We fix γ ∈ (1, 2]. It is convenient to work on the canonical space D([0, ∞), R) of cadlag paths equipped with the Skorohod distance and the corresponding Borel sigma field. We denote by X = (Xt ,t ≥ 0) the canonical process and by P the canonical distribution of a γ -stable and spectrally positive Lévy process with Laplace exponent ψ (λ ) = λ γ . Namely, E[exp(−λ Xt )] = exp(t λ γ ), for any λ ,t ≥ 0. Note that Xt is integrable and that E[Xt ] = 0, which easily implies that X oscillates when t goes to infinity. P-a.s. the path X has infinite variation (for more details, see Bertoin [5] Chapters VII and VIII ). In the more general context of spectrally positive Lévy processes, it has been proved in Le Gall and Le Jan [33] and in [12] Chapter 1 that there exists a continuous process H = (Ht ,t ≥ 0) such that for any t ≥ 0, the following limit holds in P-probability. 1 Ht := lim ε →0 ε

t 0

1{Its <Xs 0}. This allows to define H under Nγ (see the comments in Section 3.2 [12] for more details). We use the slightly abusive notation Nγ (dH) for the “distribution” of H under the excursion measure Nγ of X − I above 0.


105

For any j ∈ I , we set H j = H(a j +·)∧b j . Then the H j s are the excursions of H above 0, and the point measure

∑ δ(−Ia j ,H j )

(2.4)

j∈I

is distributed under P as Poisson point measure on [0, ∞) × D([0, ∞), R) with intensity dx ⊗ Nγ (dH). Set ζ := inf{t > 0 : Xt = 0} that is the total duration of X under Nγ . Since X does not drift to ∞, the lifetime ζ is finite Nγ -a.e. Moreover, Nγ -a.e. H0 = Hζ = 0 and Ht > 0 for any t ∈ (0, ζ ). We easily deduce from (2.3) the following scaling property for H under Nγ : For any c ∈ (0, ∞) and for any measurable function F : D([0, ∞), R) → [0, ∞), one has γ −1 1 c γ Nγ F(c− γ Hct , t ≥ 0) = Nγ F(Ht , t ≥ 0) .

(2.5)

Local Times of the Height Process We recall here from [12] Chapter 1 Section 1.3 the following result: There exists a jointly measurable process (Las , a, s ≥ 0) such that P-a.s. for any a ≥ 0, s → Las is continuous and non-decreasing and such that s 1 a dr1{a 0, let us set v(b) = Nγ (supt∈[0,ζ ] Ht > b). The continuity of H and the Poisson decomposition (2.4) obviously imply that v(b) < ∞, for any b > 0. It is moreover clear that v is non-increasing and lim∞ v = 0. For every a ∈ (0, ∞), we then define a continuous increasing process (Lta ,t ∈ [0, ζ ]), such that for every b ∈ (0, ∞) and for any t ≥ 0, one has s 1 a (2.9) lim Nγ 1{supH>b} sup dr1{a−ε 0 , v(a) = Nγ sup Ht ≥ a = Nγ Laζ = 0 = (γ −1)a γ −1 .

(2.11)

(2.12)

Lévy Trees We first define R-trees (or real trees) that are metric spaces that generalise graph-trees. Definition 2.3. Let (T, δ ) be a metric space. It is a real tree iff the following holds true for any σ1 , σ2 ∈ T . (a) There is a unique isometry fσ1 ,σ2 from [0, δ (σ1 , σ2 )] into T such that fσ1 ,σ2 (0) = σ1 and fσ1 ,σ2 (δ (σ1 , σ2 )) = σ2 . We denote by [[σ1 , σ2 ]] the geodesic joining σ1 to σ2 . Namely, [[σ1 , σ2 ]] := fσ1 ,σ2 ([0, δ (σ1 , σ2 )]). (b) If j is a continuous injective map from [0, 1] into T , such that j(0) = σ1 and j(1) = σ2 , then we have j([0, 1]) = [[σ1 , σ2 ]]. A rooted R-tree is an R-tree (T, δ ) with a distinguished point r called the root.


107

Among metric spaces, R-trees are characterized by the so-called four points inequality that is expressed as follows. Let (T, δ ) be a connected metric space. Then, (T, δ ) is a R-tree iff for any σ1 , σ2 , σ3 , σ4 ∈ T , we have δ (σ1 , σ2 )+ δ (σ3 , σ4 ) ≤ δ (σ1 , σ3 )+ δ (σ2 , σ4 ) ∨ δ (σ1 , σ4 )+ δ (σ2 , σ3 ) . (2.13) We refer to Evans [18] or to Dress, Moulton and Terhalle [9] for a detailed account on this property. The set of all compact rooted R-trees can be equipped with the pointed Gromov-Hausdorff distance in the following way. Let (T1 , δ1 , r1 ) and (T2 , δ2 , r1 ) be two compact pointed metric spaces. They can be compared one with each other thanks to the pointed Gromov-Hausdorff distance defined by dGH (T1 , T2 ) = inf δH j1 (T1 ), j2 (T2 ) ∨ δ j1 (r1 ), j2 (r2 ) . Here the infimum is taken over all (j1 , j2 , (E, δ )), where (E, δ ) is a metric space, where j1 : T1 → E and j2 : T2 → E are isometrical embeddings and where δH stands for the usual Hausdorff metric on compact subsets of (E, δ ). Obviously dGH (T1 , T2 ) only depends on the isometry classes of T1 and T2 that map r1 to r2 . In [22], Gromov proves that dGH is a metric on the set of the equivalence classes of pointed compact metric spaces that makes it a complete and separable metric space. Let us denote by T, the set of all equivalence classes of rooted compact real-trees. Evans, Pitman and Winter observed in [19] that T is dGH -closed. Therefore, (T, dGH ) is a complete separable metric space (see Theorem 2 of [19]). Let us briefly recall how R-trees can be obtained via continuous functions. We consider a continuous function h : [0, ∞) → R such that there exists a ∈ [0, ∞) such that h is constant on [a, ∞). We denote by ζh the least of such real numbers a and we view ζh as the lifetime of h. Such a continuous function is said to be a coding function. To avoid trivialities, we also assume that h is not constant. Then, for every s,t ≥ 0, we set bh (s,t) =

inf

r∈[s∧t,s∨t]

h(r) and dh (s,t) = h(s) + h(t) − 2bh(s,t).

(2.14)

Clearly dh (s,t) = dh (t, s). It is easy to check that dh satisfies the four points inequality, which implies that dh is a pseudo-metric. We then introduce the equivalence relation s ∼h t iff dh (s,t) = 0 (or equivalently iff h(s) = h(t) = bh (s,t)) and we denote by Th the quotient set [0, ζh ]/ ∼h . Standard arguments imply that dh induces a metric on Th that is also denoted by dh to simplify notation. We denote by ph : [0, ζh ] → Th the canonical projection. Since h is continuous, ph is a continuous function from [0, ζh ] equipped with the usual metric onto (Th , dh ). This implies that (Th , dh ) is a compact and connected metric space that satisfies the four points inequality. It is therefore a compact R-tree. Next observe that for any t0 ,t1 ∈ [0, ζh ] such that h(t0 ) = h(t1 ) = min h, we have ph (t0 ) = ph (t1 ); so it makes sense to define the root of (Th , dh ) by ρh = ph (t0 ). We shall refer to the rooted compact R-tree (Th , dh , ρh ) as to the tree coded by h.

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We next define the γ -stable tree as the tree coded by the γ -stable height process (Ht , 0 ≤ t ≤ ζ ) under the excursion measure Nγ and to simplify notation we set (TH , dH , ρH ) = (T , d, ρ ). We also set p = pH : [0, ζ ] → T . Note that ρ = p(0). Since Hζ = 0 and since Ht > 0, for any t ∈ (0, ζ ), ζ is the only time t ∈ [0, ζ ] distinct from 0 such that p(t) = ρ . Let us denote by T¯ the root-preserving isometry class of (T , d, ρ ). It is proved in [13] that T¯ is measurable in (T, dGH ). We then define Θγ as the “distribution” of T¯ under Nγ . Remark 2.4. We have stated the main results of the paper under Θγ because it is more natural and because Θγ has an intrinsic characterization as shown by Weill in [40]. However, each time we make explicit computations with stable trees, we have to work with random isometry classes of compact real trees, which causes technical problems (mostly measurability problems). To avoid these unnecessary complications during the intermediate steps of the proofs, we prefer to work with the specific compact rooted real tree (T , d, ρ ) coded by the γ -stable height process H under Nγ rather than to work directly under Θγ . So, we prove the results of the paper for (T , d, ρ ) under Nγ , which easily implies the same results under Θγ . The Local Time Measures and the Mass Measure on γ -Stable Trees As above mentioned, we now work with the γ -stable tree (T , d, ρ ) coded by H under the excursion measure Nγ . A certain number of definitions and ideas can be extended from graph-trees to real trees such as the degree of a vertex. Namely, for any σ ∈ T , we denote by n(σ ) the (possibly infinite) number of connected components of the open set T \{σ }. We say that n(σ ) is the degree of σ . Let σ be a vertex distinct from the root. If n(σ ) = 1, then we say that σ is a leaf of T ; if n(σ ) = 2, then we say that σ is a simple point; if n(σ ) ≥ 3, then we say that σ is a branching point of T . If n(σ ) = ∞, we then speak of σ as an infinite branching point. We denote by Lf(T ) the set leaves of T , we denote by Br(T ) the set of branching points of T and we denote by Sk(T ) = T \Lf(T ) the skeleton of T . Note that the closure of the skeleton is the whole tree Sk(T ) = T . Let us mention that H is not constant on every non-empty open subinterval of [0, ζ ], Nγ -a.e. This easily entails the following characterisation of leaves in terms of the height process: For any t ∈ (0, ζ ), p(t) ∈ Lf(T )

⇐⇒

∀ε > 0 ,

inf Hs

s∈[t−ε ,t]

and

inf Hs < Ht .

s∈[t,t+ε ]

(2.15)

Let us now define the mass measure and the local time measures on T : The mass measure m is the measure induced by the Lebesgue measure on [0, ζ ] via p. Namely, for any Borel set A of T , we have m(A) = (p−1 (A)). We can prove that the mass measure is diffuse and its topological support is clearly T . Moreover m is supported by the set of leaves: m Sk(T ) = 0. (2.16)


109

For any a ∈ (0, ∞), the a-local time measure a is the measure induced by dLa· via p. Namely, a , f =

ζ 0

dLas f (p(s)),

for any positive measurable application f on T . Let us mention that the topological support of a is included in the a-level set T (a) = {σ ∈ T : d(ρ , σ ) = a} and note from the definition that the total mass a of a is Laζ . Moreover, observe that T (a) is not empty iff sup H ≥ a. Then, (2.12) can be rewritten as follows. ∀ a > 0,

− 1 v(a) = Nγ T (a) = 0/ = Nγ (a = 0) = (γ −1)a γ −1 .

(2.17)

As already mentioned, the a-local time measure a can be defined in a purely metric way by (1.3) and there exists a modification of local time measures (a , a ≥ 0) such that a → a is Nγ -a.e. cadlag for the weak topology on the space of finite measures on T . Except in the Brownian case, a → a is not continuous and Theorem 4.7 [13] asserts that there is a one-to-one correspondence between the times of discontinuity of a → a , the infinite branching points of T and the jumps of the excursion X of the underlying γ -stable Lévy process. More precisely, a is a time-discontinuity of a → a iff there exists a (unique) infinite branching point σa ∈ T (a) such that a− = a + λa δσa . Moreover, a point σ ∈ T is an infinite branching point iff there exists t ∈ [0, ζ ] such that p(t) = σ and Δ Xt > 0; if furthermore σ = σa , then λa = Δ Xt . Now, observe that if σ ∈ T (a) is an atom of a , the definition (1.3) of a entails that σ is an infinite branching point and that a is a time-discontinuity of a → a . Thus, σ = σa . Recall that the Ray-Knight theorem for H asserts that a → a is distributed as a CSBP (under its excursion measure), which has no fixed time-discontinuity. This (roughly) explains the following. ∀ a > 0,

Nγ − a.e. a is diffuse.

(2.18)

We refer to [13] for more details. The Branching Property for H We now describe the distribution of excursions of the height process above level b (or equivalently of the corresponding stable tree above level b). Let us fix b ∈ (0, ∞), and denote by (gbj , d bj ), j ∈ Ib , the connected components of the open set {t ≥ 0 : Ht > b}. For any j ∈ Ib , denote by H b, j the corresponding excursion of H that defined by Hsb, j = H(gb +s)∧d b − b, s ≥ 0. j j This decomposition is interpreted in terms of the tree as follows. Recall that ¯ ρ , b) stands for the closed ball with center ρ and radius b. Observe that the conB( ¯ ρ , b) are the subtrees T˜j b,o := p((gbj , d bj )), nected components of the open set T \B( j ∈ Jb . The closure T j b of T j b,o is simply {σ jb } ∪ T j b,o , where σ jb = p(gbj ) = p(d bj ), that is the points on the b-level set T (b) at which T j b,o is grafted. Observe that the rooted compact R-tree (T j b , d, σ jb ) is isometric to the tree coded by H b, j .

110

T. Duquesne

We then define H˜ sb = Hτsb , where for every s ≥ 0, we have set t τsb = inf t ≥ 0 : ds 1{Hs ≤b} > s . 0

The process H˜ b = (H˜ sb , s ≥ 0) is the height process below b and the rooted compact ¯ ρ , b), d, ρ ) is isometric to the tree coded by H˜ b . Let Gb be the sigmaR-tree (B( field generated by H˜ b augmented by the Nγ -negligible sets. From the approximation (2.9), it follows that Lbζ is measurable with respect to Gb . We next use the following notation Nγ(b) = Nγ ( · | sup H > b)

(2.19)

that is a probability measure and we define the following point measure on [0, ∞) × D([0, ∞), R): Mb = ∑ δ(Lb ,H b, j ) . (2.20) j∈Ib

gbj

The branching property at level b then asserts that under Nγ(b) , conditionally given Gb , Mb is distributed as a Poisson point measure with intensity 1[0,Lb ] (x)dx ⊗ ζ

Nγ (dH). We refer to Proposition 1.3.1 in [12] or to the proof of Proposition 4.2.3 [12]. Let us mention that it is possible to rewrite intrinsically the branching property under Θγ : we refer to Theorem 4.2 [13] for more details. Spinal Decomposition at a Random Time We recall another decomposition of the height process (and therefore of the corresponding tree) that is proved in [12] Chapter 2 and in [13] under a more explicit form (see also [15] for further applications). This decomposition is used in a crucial way in the proof of the upper- and lower-density results for the local times measures and the mass measure. Let us introduce an auxiliary probability space (Ω , F , P) that is assumed to be rich enough to carry the various independent random variables we shall need. Let (Ut ,t ≥ 0) be a subordinator defined on (Ω , F , P) with initial value U0 = 0 and with Laplace exponent ψ (λ ) = γλ γ −1 , λ ≥ 0. Let N ∗=

∑

j∈I ∗

δ(r∗j , H ∗ j )

(2.21)

be a random point measure on [0, ∞) × D([0, ∞), R) defined on (Ω , F , P) such that a regular version of the law of N ∗ conditionally given U is that of a Poisson point measure with intensity dUr ⊗ Nγ (dH). Here dUr stands for the (random) Stieltjes measure associated with the non-decreasing path r → Ur . For any a ∈ (0, ∞), we also set Na∗ = ∑ 1[0,a] (r∗j ) δ(r∗ , H ∗ j ) . (2.22) j∈I ∗

j


111

We next consider the γ -height process H under its excursion measure Nγ . For any t ≥ 0, we set Hˆ t := (H(t−s)+ , s ≥ 0) (here, ( ·)+ stands for the positive part function) and Hˇ t := (H(t+s)∧ζ , s ≥ 0). We also define the random point measure Nt on [0, ∞)× D([0, ∞), R) by Nt = N (Hˆ t ) + N (Hˇ t ) := ∑ δ(rt ,H ∗ t, j ) , (2.23) j

j∈Jt

where for any continuous function h : [0, ∞) → [0, ∞) with compact support, the point measure N (h) is defined as follows: Set h(t) = inf[0,t] h and denote by (gi , di ), i ∈ I (h) the excursion intervals of h − h away from 0 that are the connected component of the open set {t ≥ 0 : h(t) − h(t) > 0}. For any i ∈ I (h), set hi (s) = ((h − h)((gi + s) ∧ di ) , s ≥ 0). We then define N (h) as the point measure on [0, ∞) × D([0, ∞), R) given by N (h) =

∑

i∈I (h)

δ(h(gi ),hi ) .

Lemma 3.4 in [13] asserts the following. For any a and for any nonnegative measurable function F on the set of positive measures on [0, ∞) × D([0, ∞), R) (equipped with the topology of vague convergence), one has Nγ

ζ

0

dLta

F Nt = E [F(Na∗ )] .

(2.24)

We shall refer to this identity as to the spinal decomposition of H at a random time. Let us briefly interpret this decomposition in terms of the γ -stable tree T coded by H. Choose t ∈ (0, ζ ) and set σ = p(t) ∈ T . Then the geodesic [[ρ , σ ]] is interpreted as the ancestral line of σ . Let us denote by T j o , j ∈ J , the connected components of the open set T \[[ρ , σ ]] and denote by T j the closure of T j o . Then, there exists a point σ j ∈ [[ρ , σ ]] such that T j = {σ j }∪T j o . Recall notation (rtj , H ∗ t, j ), j ∈ Jt from (2.23). The specific coding of T by H entails that for any j ∈ J there exists a unique j ∈ Jt such that d(ρ , σ j ) = rtj and such that the rooted compact R-tree (T j , d, σ j ) is isometric to the tree coded by H ∗ t, j ¯ We now compute m(B(p(t), r)) in terms of Nt as follows. First, recall from (2.14) the definition of b(s,t) and d(s,t). Note that if Hs = b(s,t) with s = t, then p(s) ∈ Sk(T ) by (2.15). Let us fix a radius r in (0, Ht ). Then, (2.16) entails ¯ m B(p(t), r) =

ζ 0

1{d(s,t)≤r} ds =

ζ 0

1{0 0. Recall the definition of the a-local time measure a (whose total mass a is equal to Laζ ) and recall that a −λ La Nγ 1 − e−λ = Nγ 1 − e ζ = u(a, λ ) = (γ −1)a +

1

λ γ −1

−

1 γ −1

.

(2.31)

Next recall from (2.19) the definition of Nγ(a) . We easily deduce from (2.12) and (2.31) that (γ −1)aλ γ −1 1 γ −1 . (2.32) Nγ(a) exp(−λ a ) = 1 − 1 + (γ −1)aλ γ −1 Consequently, a

1 − γ −1

(law)

a under Nγ(a) = 1 under Nγ(1) .

(2.33)

Lemma 2.5. For any γ ∈ (1, 2], we have Nγ(1) 1 ≤ x ∼x→0+

xγ −1 . (γ −1)2Γ (γ )

Proof. From (2.32), we get λ −(γ −1) Nγ(1) exp(−λ 1 ) ∼λ →∞ . (γ − 1)2 The desired result is then a direct consequence of a Tauberian theorem due to Feller: see [20] Chapter XIII 5 (see also [8] Theorem 1.7.1’, p. 38). Recall the notation N r ≤ 2a, we set

∗

and the definition of L∗r (a) from (2.29). For any 0 ≤ r ≤

Λr ,r (a) =

a−r∗

∑∗ 1[a− 2r , a− r2 )(r∗j ) Lζ ∗j j .

(2.34)

j∈I

Observe that ∀ 0 ≤ r ≤ r ≤ 2a ,

L∗r (a) = Λr ,r (a) + L∗r (a).

(2.35)

Lemma 2.6. Let (rn , n ≥ 0) be a sequence such that 0 < rn+1 ≤ rn ≤ 2a and limn rn = 0. Then, the random variables (Λrn+1 ,rn (a), n ≥ 0) are independent and L∗r0 (a) =

∑ Λrn+1 ,rn (a).

n≥0

(2.36)

114

T. Duquesne

Proof. First, note that (2.36) is a direct consequence of the definitions of Λr ,r (a) and of L∗r (a). Let us prove the independence property. Recall that conditionally given U, N ∗ is a Poisson point process with intensity dUt ⊗ Nγ (dH). Elementary properties of Poisson point processes and the definition of the Λrn+1 ,rn (a)s entail that the random variables (Λrn+1 ,rn (a), n ≥ 0) are independent conditionally given U. Moreover, the conditional distribution of Λrn+1 ,rn (a) given U only involves the increments of U on [rn+1 , rn ], which easily implies the desired result since U is a subordinator. Remark 2.7. The previous lemma and (2.35) imply that for any 0 ≤ r ≤ r ≤ 2a, one has L∗r (a) ≥ Λr ,r (a) and that Λr ,r (a) is independent of L∗r (a). Observe also that the process r → L∗r (a) has independent increments. Lemma 2.8. For any 0 ≤ r ≤ r ≤ 2a, we have E exp(−λ Λr ,r (a)) =

γ −1 2 γ −1 2

r λ γ −1 + 1

γ γ −1

rλ γ −1 + 1

.

(2.37)

Consequently, we get r

1 − γ −1

(law)

Λr ,r (a) = Λ r ,1 (1). r

Proof. First observe that the second point is an immediate consequence of the first one. Recall that conditionally given U, N ∗ is distributed as a Poisson point process with intensity dUt ⊗ Nγ . Therefore, E exp − λ Λr ,r (a) |U = exp −

−λ La−t . dUt Nγ 1 − e ζ

[a−r/2,a−r /2 )

−λ La−t Recall that u(a − t, λ ) = Nγ 1 − e ζ , where u is given by (1.1) and recall that U is a subordinator with Laplace exponent λ → γλ γ −1 . Thus, E exp − λ Λr ,r (a) = exp − γ

u(a − t, λ )γ −1dt ,

a−r /2 a−r/2

which entails the desired result thanks to a simple change of variable. Taking r = 0 in the previous lemma entails the following. Lemma 2.9. For any a ∈ (0, ∞) and for any r ∈ [0, 2a], we have − γ γ −1 E exp(−λ L∗r (a)) = 1 + 2 rλ γ −1 γ −1 . Then, r−1/(γ −1) L∗r (a) has the same law as L∗1 (1).


115

To simplify notation, we set Zγ := L∗1 (1) and Zγ := Λ 1 ,1 .

(2.38)

2

Proposition 2.10. We have the following estimates. •

(i) For γ ∈ (1, 2), we have lim xγ −1 P(Zγ ≥ x) = 2 lim xγ −1 P(Zγ ≥ x) =

x→∞

•

x→∞

γ . 2Γ (2 − γ )

(ii) For any γ ∈ (1, 2] we get γ

lim x

−γ

x→0+

P(Zγ ≤ x) =

2 γ −1 γ

(γ − 1) γ −1 Γ (1 + γ )

.

Proof. First assume that γ ∈ (1, 2). When λ goes to 0, we have γ γ E e−λ Zγ = 1 − 2 λ γ −1 + o(λ γ −1) and E e−λ Zγ = 1 − 4 λ γ −1 + o(λ γ −1). A Tauberian theorem due to Bingham and Doney [7] (see also [8] Theorem 8.1.6, p. 333) implies (i). Let us prove (ii). We have γ ∈ (1, 2]. When λ goes to ∞, we get

γ

lim λ E e

λ →∞

−λ Zγ

γ

=

2 γ −1 γ

(γ − 1) γ −1

.

Then, (ii) is a consequence of a Tauberian theorem due to Feller ([20] Chapter XIII, 5; see also [8] Theorem 1.7.1’, p. 38). Recall the definition of Mr∗ (a) from (2.26). For any 0 ≤ r ≤ r ≤ a, we set Qr ,r (a) =

∑

j∈I ∗

1[a−r,a−r ) (r∗j )

ζ∗ j 0

1{H ∗ j ≤r−a+r∗ } . s

(2.39)

j

Arguing as in Lemma 2.6, we prove the following independence property. Lemma 2.11. Let (rn , n ≥ 0) be a sequence such that 0 < rn+1 ≤ rn ≤ a and limn rn = 0. Then, the random variables (Qrn+1 ,rn (a), n ≥ 0) are independent. Remark 2.12. Note that the increments of r ∈ [0, a] → Mr∗ (a) are not independent. However, for any 0 ≤ r ≤ r ≤ a, we have Mr∗ (a) − Mr∗ (a) = Qr ,r (a) +

∑∗ 1[a−r , a](r∗j )

j∈I

ζ∗ j 0

1{r −a+r∗ 0, one has 1

γ

κa (λ , 0) = c γ −1 κc a (c− γ −1 λ , 0), which easily implies the second point of the lemma. Take r = 0 in the previous lemma to get the following lemma. Lemma 2.14. For any a ∈ (0, ∞) and for any r ∈ [0, a], we have κr (λ , 0)γ . E exp(−λ Mr∗ (a)) = 1 − λ Then, r−γ /(γ −1) Mr∗ (a) has the same law as M1∗ (1). To simplify notation, let us set Yγ := M1∗ (1).


117

Proposition 2.15. For any γ ∈ (1, 2), we have lim xγ −1 P(Yγ ≥ x) =

x→∞

1 . Γ (2 − γ )

Proof. Recall (2.10), recall that 0ζ 1{Hs ≤a} ds = for any b ∈ (0, ∞) (see (2.11)). Thus, κa (λ , 0) =N λ λ →0

ζ

lim

0

a 0

Lbζ db and recall that N(Lbζ ) = 1,

a 1{Hs ≤a} ds = N Lbζ db = a.

(2.41)

0

Take a = 1 in (2.41) and use Lemma 2.14 to get E e−λ Yγ = 1 − λ γ −1 + o(λ γ −1) when λ goes to 0. Since 0 < γ − 1 < 1, a Tauberian theorem due to Bingham and Doney [7] entails the desired result (see also [8] Theorem 8.1.6, p. 333).

3 Proofs of the Main Results 3.1 Proof of Theorem 1.1 Let us fix a ∈ (0, ∞) and let g : (0, 1) → (0, ∞) be such that lim0+ r−1/(γ −1) g(r) = 0. To simplify notation we set h(r) = r−1/(γ −1) g(r). Lemma 2.9 and Proposition 2.10 (ii) imply that for all sufficiently large n, P(L∗2−n (a) ≤ g(2−n )) = P(Zγ ≤ h(2−n )) ∼n→∞ Kγ h(2−n )γ ,

(3.1)

where Kγ is the limit on the right member of Proposition 2.10 (ii). We first prove Theorem 1.1 (i). So we assume

∑ h(2−n)γ < ∞.

(3.2)

n≥1

Borel-Cantelli and (3.1) imply P(lim infn→∞ L∗2−n (a)/g(2−n) ≥ 1) = 1. This easily entails P(lim infn→∞ L∗2−n (a)/g(2−n) = ∞) = 1, since (3.2) is also satisfied by K · h for arbitrarily large K. Then, (2.30) implies Nγ

T

a (d σ )1{lim infn a (B(σ ,2−n ))/g(2−n) b} and recall that H b, j is the corresponding excursion of H above b corresponding to (gbj , d bj ). We set T j b = p([gbj , d bj ]) and σ jb = p(gbj ) = p(d bj ). As already mentioned (T j b , d, σ jb ) is isometric to the tree coded by H b, j . The total height of T j b is then Γ (T j b ) = sups≥0 Hsb, j . For any η > 0, we set Db,η = {T jb ; i ∈ Ib : Γ (T j b ) > η }. Note that Db,η is a finite set. Observe that a (T j b ) = Lad b − Lagb is the local time at j

j

level a − b of H b, j , or equivalently the total mass of the local time measure at level a − b of T j b . Then, the branching property entails for any x > 0, Nγ(b)

∑

1{a (T )≤x} Gb = Lbζ Nγ La−b ≤ x ; sup H > a − b . ζ

T ∈Db,a−b

Recall that La−b = a−b . Then, (2.12) and the scaling property (2.33) imply ζ − 1 − 1 ≤ x ; sup H > a − b = (γ −1)(a−b) γ −1 Nγ(1) 1 ≤ (a−b) γ −1 x . Nγ La−b ζ Recall that Nγ -a.e. Lbζ = b = 0, on {sup H ≤ b}. Thus, (2.11) and (2.12) entail Nγ

∑

− 1 1 1{a (T )≤x} = (γ −1)(a−b) γ −1 Nγ(1) 1 ≤ (a−b)− γ −1 x .

(3.3)

T ∈Db,a−b

For any n ∈ N such that 2−n < a, we next set Vn =

∑

T ∈Da−2−n ,2−n

g(2 · 2−n)1{a (T )≤g(2·2−n)} .

We apply (3.3) with b = a − 2−n and η = 2−n , and we use Lemma 2.5 to get 1 − γ −1

N(Vn ) = (γ − 1)

n − 1 2 γ −1 g(2 · 2−n)Nγ(1) 1 ≤ 2 γ −1 h(2 · 2−n) ≤ Kγ h(2 · 2−n)γ ,

where Kγ is a positive constant that only depends on γ . Therefore, (3.2) entails Nγ − a.e.

lim

n→∞

∑ Vp = 0.

p≥n

(3.4)


119

¯ σm , rm ) ; m ≥ 1) be any Let ε ∈ (0, a/2). We assume that T (a) = 0. / Let (B( ¯ σm , rm ) are pairwise disjoint, ε -closed packing of E. Namely, the closed balls B( σm ∈ E ⊂ T (a) and rm ≤ ε , for any m ≥ 1. Let us fix m ≥ 1. There exists n (that depends on m) such that 2−n < rm ≤ 2 · 2−n . Now observe that T (a) is the union of the sets T ∩ T (a) where T ranges in Da−2−n−1 ,2−n−1 . Consequently, there exists T ∗ ∈ Da−2−n−1 ,2−n−1 such that σm ∈ T ∗ ∩ T (a). Denote by σ ∗ , the lowest point in T ∗ . Namely σ ∗ is the point of T ∗ that is the closest to the root and σ ∗ ∈ T (a − 2−n−1). It is easy to prove that for any σ ∈ T ∗ ∩ T (a), we have d(σ , σm ) ≤ d(σ , σ ∗ ) + d(σ , σ ∗ ) = 2 · 2−n−1 = 2−n < rm . ¯ σm , rm ). Thus, a (T ∗ ) ≤ a (B( ¯ σm , rm )). Since this Thus T ∗ ∩ T (a) ⊂ T (a) ∩ B( holds true for any m ≥ 1, we get

∑ g(rm )1{a(B(σm ,rm ))0}

= 0.

a

Nγ

T

This proves (1.10) in Theorem 1.1 (ii). Furthermore, if g is a regular gauge function, / then, (1.10) and Lemma 2.1 (i) entail that Nγ -a.e. Pg (T (a) ) = ∞, on {T (a) = 0}, which completes the proof of Theorem 1.1.

3.2 Proof of Proposition 1.5 Fix a > 0 and let g be as in Proposition 1.5. Namely g : (0, 1) → (0, ∞) is such that lim0+ r−1/(γ −1) g(r) = ∞. To simplify notation we set h(r) = r−1/(γ −1) g(r). Although Proposition 1.5 (i) is already proved in [14] we provide a brief proof of it: We assume that (3.10) ∑ h(2−n)−(γ −1) < ∞. n≥1

The scaling property stated in Lemma 2.9 and Proposition 2.10 (i) imply that P(L∗2−n (a) ≥ g(2−n )) = P(Zγ ≥ h(2−n )) ∼n→∞

γ h(2−n)−(γ −1) . 2Γ (2 − γ )

Borel-Cantelli entails P(lim supn→∞ L∗2−n (a)/g(2−n ) ≤ 1). Since (3.10) is satisfied by c · h for arbitrarily small c > 0, we easily get P(lim supn→∞ L∗2−n (a)/g(2−n) = 0) = 1 and (2.30) entails Nγ

T

a (d σ )1{lim supn a (B(σ ,2−n ))/g(2−n)>0}

= 0.

122

T. Duquesne

This proves (1.13) in Proposition 1.5 (i). Furthermore, if g is a regular gauge function, then, (1.13) and Lemma 2.1 (iii) entail that Nγ -a.e. Hg (T (a) ) = ∞, on {T (a) = 0}, / which completes the proof of Proposition 1.5 (i). Let us prove (1.14) in Proposition 1.5 (ii). We now assume

∑ h(2−n)−(γ −1) = ∞.

(3.11)

n≥1

For any n ≥ 1, set εn = 1{Λ −n−1 2

and Proposition 2.10 (i) imply

,2−n

E[εn ] ∼n→∞

(a) ≥ g(2−n )} .

The scaling property in Lemma 2.8

γ h(2−n )−(γ −1) . 4Γ (2 − γ )

Therefore ∑n≥1 E[εn ] = ∞. The independence property stated in Lemma 2.6 shows that the εn ’s are independent. The converse of Borel-Cantelli implies P(∑n≥1 εn = ∞) = 1. As noticed in Remark 2.7, we have εn ≤ 1{L∗−n (a)≥g(2−n)} . Consequently, 2 P(lim supn→∞ L∗2−n (a)/g(2−n ) ≥ 1) = 1. Since (3.11) is satisfies by K · h for arbitrarily large K, we easily get P(lim supn→∞ L∗2−n (a)/g(2−n) = ∞) = 1 and (2.30) entails Nγ

(d σ )1{lim supn a (B(σ ,2−n))/g(2−n ) 0. The scaling property stated in Lemma 2.14 and Proposition 2.15 (i) imply that P(M2∗−n (a) ≥ g(2−n )) = P(Yγ ≥ h(2−n )) ∼n→∞

h(2−n )−(γ −1) . Γ (2 − γ )

Borel-Cantelli implies P(lim supn M2∗−n (a)/g(2−n) ≤ 1) = 1. Since (3.12) is satisfied by c · h for arbitrarily small c > 0, we easily get P(lim supn→∞ M2∗−n (a)/g(2−n ) = 0) = 1. By (2.27), for any a > 0, we get


Nγ

123

T

a (d σ )1{lim supn m(B(σ ,2−n))/g(2−n )>0}

= 0.

Since m = 0∞ a , this entails (1.16) in Proposition 1.9. Furthermore, if g is a regular gauge function, then, (1.16) and Lemma 2.1 (iii) imply that Hg (T ) = ∞, Nγ -a.e., which completes the proof of Proposition 1.9 (i). Let us prove (1.17) in Proposition 1.9 (ii). We assume

∑ h(2−n)−(γ −1) = ∞.

(3.13)

n≥1

For any n ≥ 1, we set εn = 1{Q −n−1 2

,2−n

(a) ≥ g(2−n )} .

The scaling property stated in

Lemma 2.13 and Proposition 2.15 (i) entail −γ 2γ h(2−n )−(γ −1) E[εn ] = P Yγ ≥ 2 γ −1 h(2−n ) ∼n→∞ , Γ (2 − γ )

Thus, ∑n≥1 E[εn ] = ∞. The independence property of Lemma 2.11 (i) implies that the εn ’s are independent. Thus, P(∑n≥1 εn = ∞) = 1, by the converse of Borel-Cantelli. Then, Remark 2.12 entails εn ≤ 1{M∗−n (a)≥g(2−n)} , for any n ≥ 1. 2

Thus, P(lim supn→∞ M2∗−n (a)/g(2−n) ≥ 1) = 1. Since (3.13) is satisfied by K · h for arbitrarily large K, we easily get P(lim supn→∞ M2∗−n (a)/g(2−n) = ∞) = 1. By (2.27), for any a > 0, we get Nγ

Since m =

(d σ )1{lim supn m(B(σ ,2−n ))/g(2−n) 0. Recall the definition of H˜ a that is the height process below a and recall that Ga is the sigma-field generated by H˜ a augmented with the Nγ -negligible

sets. We denote by Ga− the sigma-field generated by b 1/(γ − 1), then Hg (T (a) ) = 0, Nγ -a.e. and if q < 1/(γ − 1), then Hg (T (a) ) = ∞, Nγ -a.e. on {T (a) = 0}. / We then restrict our attention to the case q = 1/(γ − 1). The general idea of the proof of Theorem 1.6 is the following: if for a certain a ∈ (0, ∞), we have Nγ ( 0 < Hg (T (a)) < ∞) > 0, then we first prove that / We next observe that Hg (· ∩ T (a) ) 0 < Hg (T (a)) < ∞, Nγ -a.e. on {T (a) = 0}. behaves like a with respect to the scaling property and the branching property and we prove it entails that Hg (· ∩ T (a) ) = c0 a , where c0 ∈ (0, ∞). Finally, we get a contradiction thanks to the test stated in Proposition 1.5. The proof is in several steps. We first discuss how a → Hg (T (a)) ∈ [0, ∞] behaves with respect to the branching property. We agree on the convention exp(−∞) = 0. Then, for any a, λ ∈ (0, ∞), it makes sense to set u(a, ˜ λ ) = Nγ 1 − e−λ Hg(T (a) ) , Recall from (2.12) the definition of v(a) and observe that u(a, ˜ λ ) ≤ v(a) < ∞. Let us fix b ∈ (0, a). Recall that (gbj , d bj ), j ∈ Ib stands for the connected components of the open set {t ≥ 0 : Ht > b} and recall that for any j ∈ Ib , we denote by H b, j the corresponding excursion of H above b. We also set T j b = p([gbj , d bj ]) and σ jb = p(gbj ). Then, the subtree (T j b , d, σ jb ) is isometric to the rooted compact real tree coded by the excursion H b, j . For any n ≥ 1, we set Ln = ∑ j∈Ib n ∧ Hg (T j b (a − b) ), where T j b (a − b) := {σ ∈ T j b : d(σ jb , σ ) = a − b}. Note that T j b (a − b) is the (a − b)-level set of T j b . Since Hg (T j b (a − b) ) is a measurable function of H b, j , the branching property (2.20) applies and for any λ ∈ (0, ∞), we Nγ(b) -a.s. get −Lb u˜ (a−b,λ ) Nγ(b) e−λ Ln Gb = e ζ n , where uñ (a − b, λ ) = Nγ (1 − exp(−λ n ∧ Hg (T (a − b) )) ). By monotone convergence, we get limn uñ (a − b, λ ) = u(a ˜ − b, λ ). Then, observe that limn ↑ Ln =


125

Hg (T (a) ). Thus, the conditional dominated convergence theorem implies that for any λ ∈ (0, ∞), we Nγ(b) -a.s. have −Lb u(a−b, ˜ λ) Nγ(b) e−λ Hg(T (a) ) Gb = e ζ .

(3.14)

Since Lbζ = 0, Nγ -a.e. on {sup H ≤ b}, this entails −Lb u(a−b, ˜ λ) = u(b, u(a ˜ − b, λ ) ). u(a, ˜ λ ) = Nγ 1 − e ζ

(3.15)

Note that Theorem 1.6 is implied by the two following claims. (Claim 1) If there exists a0 ∈ (0, ∞) such that Nγ (Hg (T (a0 ) ) = ∞) > 0, then for any a ∈ (0, ∞), Nγ -a.e. Hg (T (a)) = ∞, on {T (a) = 0}. / (Claim 2) For any a ∈ (0, ∞), Nγ (0 < Hg (T (a)) < ∞) = 0. ˜ λ) = We first prove (Claim 1). To that end, observe that u(b, ˜ 0+) = limλ →0 u(b, Nγ (H (T (b)) = ∞) for any b ∈ (0, ∞). Then, (3.15) entails u(a, ˜ 0+) = u(a − b, u(b, ˜ 0+) ) ,

a > b > 0.

(3.16)

Let us now recall the scaling property of T : Let c ∈ (0, ∞). The “law” of (T , cd, ρ ) under Nγ is the “law” of (T , d, ρ ) under c1/(γ −1) Nγ . We next denote by Hg,cd the g-Hausdorff measure on (T , cd, ρ ) and we set gc (r) = g(cr), for any r ∈ (0, 1). Then, for any b > 0, we easily get Hg,c d {σ ∈ T : c d(ρ , σ ) = b} = Hgc {σ ∈ T : d(ρ , σ ) = b/c} 1 = c γ −1 Hg T (b/c) , since g is regularly varying at 0 with exponent 1/(γ − 1). The scaling property for T implies that the law of c1/(γ −1) Hg (T (b/c)) under Nγ is the same as the law of Hg (T (b)) under c1/(γ −1) Nγ . Thus u(b, ˜ 0+) = b−1/(γ −1) u(1, ˜ 0+), for any b > 0. If there exists a0 > 0 such that Nγ (Hg (T (a0 )) = ∞) > 0, then u(1, ˜ 0+) > 0, ˜ 0+) = ∞. Recall that limb→0 u(b, / = lim u(a, μ ), v(a) = Nγ (sup H > a) = Nγ (Laζ > 0) = Nγ (T (a) = 0) μ →∞

where u is given by (1.1). Then, (3.16) and the previous arguments easily im˜ 0+) ) = v(a). Namely, ply that for any a ∈ (0, ∞), u(a, ˜ 0+) = limb→0 u(a − b, u(b, / Since {H (T (a)) = ∞} ⊂ {T (a) = 0}, / it Nγ (H (T (a)) = ∞) = Nγ (T (a) = 0). implies that Nγ -a.e. Hg (T (a) ) = ∞, on {T (a) = 0}, / which proves the first claim. To prove (Claim 2), we argue by contradiction and we suppose that there exists a0 ∈ (0, ∞) such that Nγ (0 < Hg (T (a0 ) ) < ∞) > 0. (3.17)

126

T. Duquesne

The previous arguments show that u(b, ˜ 0+) = Nγ (H (T (b) ) = ∞) = 0, for any b ∈ (0, ∞). The scaling property discussed above entails u(b, ˜ λ) = c

1 − γ −1

1 u˜ b/c , c γ −1 λ ,

b, λ , c > 0.

(3.18)

Nγ (H (T (a) ) > 0) = Nγ (sup H > a) = v(a).

(3.19)

We first claim that for any a ∈ (0, ∞),

˜ λ ) = Nγ (H (T (b) ) > 0). Then, (3.18) Indeed, observe that v(b) ˜ := limλ →∞ ↑ u(b, implies v(b) ˜ = b−1/(γ −1) v(1). ˜ Assumption (3.17) entails that 0 < u(a ˜ 0 , λ ) ≤ v(a ˜ 0 ). ˜ = ∞. Thanks to (3.15), we get Thus, we get v(1) ˜ > 0, which implies limb→0 v(b) v(a) ˜ = u(a − b, v˜(b) ) and v(a) ˜ = limb→0 u(a − b, v(b) ˜ ) = v(a), which is (3.19). Recall that for any fixed λ ∈ (0, ∞), b → u(b, λ ) is decreasing. Then, (3.15) ˜ λ ), for any a > b > 0, and for any λ > 0. Thus, it implies that u(a, ˜ λ ) ≤ u(b, makes sense to set φ (λ ) = limb↓0 ↑ u(b, ˜ λ ) ∈ (0, ∞]. Then (3.15) entails u(a, ˜ λ) = u(a, φ (λ )), for any a, λ > 0, with the convention: u(a, ∞) = v(a). Since we have Nγ (Hg (T (a)) = ∞) = 0, (3.19) and the definition of u˜ imply u(a, ˜ λ ) < v(a). Consequently, φ (λ ) ∈ (0, ∞), for any λ > 0. Next, observe that u satisfies the same scaling property (3.18) as u. ˜ Therefore, c1/(γ −1) φ (λ ) = φ (c1/(γ −1) λ ), for any c, λ > 0. Namely, φ (λ ) = c0 λ , where c0 := φ (1) ∈ (0, ∞) and we have proved that u(b, ˜ λ ) = u(b, c0 λ ),

λ , b > 0.

(3.20)

We next prove that for any a > b > 0, and for any λ ≥ 0, −λ c La Nγ(a) − a.s. Nγ(a) e−λ Hg (T (a) ) Gb = Nγ(a) e 0 ζ Gb .

(3.21)

−λ c La Proof of (3.21). By the branching property, we easily get Nγ(b) e 0 ζ Gb = e

−Lbζ u(a−b,c0 λ )

. Therefore, −λ c La Nγ(b) e−λ Hg(T (a) ) Gb = Nγ(b) e 0 ζ Gb .

Then, we get Nγ(b) (1{Hg (T (a) )=0} |Gb ) = Nγ(b) (1{La =0} |Gb ) = e ζ

(b)

go to ∞. Thus, Nγ -a.s.

−Lbζ v(a−b)

, by letting λ

−λ c La Nγ(b) 1{Hg (T (a) )>0} e−λ Hg(T (a) ) Gb = Nγ(b) 1{La >0} e 0 ζ Gb . ζ

By (2.12) and (3.19), we have 1{La >0} = 1{Hg (T (a) )>0} = 1{supH>a} , Nγ -a.e. Thus, Nγ(b) -a.s. we get

ζ

−λ c La Nγ(b) 1{sup H>a} e−λ Hg(T (a) ) Gb = Nγ(b) 1{sup H>a} e 0 ζ Gb .


127

Recall that Nγ(b) = Nγ (· ∩ {sup H >b})/v(b) and note that {sup H >a}⊂{supH >b}. Thus, for any positive Gb -measurable random variable Y , we get −λ c La Nγ 1{sup H>a} e−λ Hg(T (a) )Y = Nγ 1{supH>a} e 0 ζ Y ,

which easily entails (3.21).

Recall that Laζ and Hg (T (a) ) are Ga -measurable and recall that Ga− = Ga . By letting b go to a in (3.21), we get Hg (T (a) ) = c0 Laζ , Nγ(a) -a.s. which easily entails Hg (T (a) ) = c0 Laζ , Nγ -a.e. Recall that a (T (a) ) = Laζ . Thus, we have proved: ∀a ∈ (0, ∞) ,

Nγ − a.e. Hg (T (a) ) = c0 a (T (a) ).

(3.22)

We now prove the following. Nγ − a.e. Hg ( · ∩ T (a) ) = c0 a .

(3.23)

Proof of (3.23). Let a > b > 0. The branching property and (3.22) easily imply that Nγ -a.e. for any j ∈ Ib , we have Hg (T j b (a − b) ) = c0 a (T j b (a − b) ). For any σ ∈ T (a) and any r ≥ 0, we set B¯ a (σ , r) = {σ ∈ T (a) : d(σ , σ ) ≤ r}. Note that B¯ a (σ , r) is the closed ball in (T (a), d) with radius r and center σ . Let us fix σ ∈ T (a) and let us denote by σ˜ the unique point in [[ρ , σ ]] such that d(σ , σ˜ ) = a − b. Observe that B¯ a (σ , 2(a − b)) is the union of the T j b (a − b) such that σ jb = σ˜ . Since the T j b (a − b)s are pairwise disjoints, we get ∀a > b > 0, Nγ − a.e. ∀σ ∈ T (a), H (B¯ a (σ , 2(a−b)) ) = c0 a (B¯ a (σ , 2(a−b)) ). Consequently, there exists a Borel subset A ⊂ D([0, ∞), R) whose complementary set is Nγ -negligible and such that on A, one has Hg (B¯ a (σ , r)) = c0 a (B¯ a (σ , r)) < ∞, for any σ ∈ T (a) and any r ∈ Q+ , which easily entails (3.23). We have proved that (3.17) implies that there exists c0 ∈ (0, ∞) such that (3.23) holds true. Let us furthermore assume that 2−n

∑ g(2−n)γ −1

< ∞.

(3.24)

n≥1

Then, Proposition 1.5 (i) implies that Hg (T (a)) = ∞, Nγ -a.e. on {T (a) = 0}, / which contradict (3.23) since Nγ (a (T (a)) = ∞) = 0. Consequently (3.24) fails and Proposition 1.5 (ii) entails that a (E) = a (T (a)), where E is a Borel subset of T (a) such that lim supn a (B(σ , 2−n ))/g(2−n ) = ∞ for any σ ∈ E. By the comparison lemma for Hausdorff measures (Lemma 2.1 (iv)), we get Hg (E) = 0, which contradicts (3.23). This implies that (3.17) is false, which proves (Claim 2). This completes the proof of Theorem 1.6.

128

T. Duquesne

Proof of Theorem 1.10 Let g(r) = rq s(r) where q is nonnegative and s is slowly varying at 0. Recall that we furthermore assume that g is a regular gauge function. Recall that Nγ -a.e. the Hausdorff dimension of T is γ /(γ − 1). Thus, if q > γ /(γ − 1), then Hg (T ) = 0, Nγ -a.e. and if q < γ /(γ − 1), then Hg (T ) = ∞, Nγ -a.e. We next restrict our attention to the case q = γ /(γ − 1). The general idea of the proof of Theorem 1.10 is the following: if Nγ ( 0 < Hg (T ) < ∞) > 0, then we first prove that 0 < Hg (T ) < ∞, Nγ -a.e. We next observe that Hg behaves like m with respect to the scaling property and the branching property and we prove it entails that Hg = c0 m, where c0 ∈ (0, ∞). Finally, we get a contradiction thanks to the test stated in Proposition 1.9. We first need to state two preliminary results. We agree on the convention exp(−∞) = 0 and for any a, λ > 0 and any μ ≥ 0, we set ¯ ρ ,a) )− μ La −λ Hg(B( ζ ∈ [0, ∞]. κ˜ a (λ , μ ) = Nγ 1 − e Let us fix b ∈ (0, a). Recall that (gbj , d bj ), j ∈ Ib stands for the connected components of the open set {t ≥ 0 : Ht > b} and recall that for any j ∈ Ib , we denote by H b, j the corresponding excursion of H above b. We also set T j b = p([gbj , d bj ]) and σ jb = p(gbj ). Then, the subtree (T j b , d, σ jb ) is isometric to the rooted compact real tree coded by the excursion H b, j . We set T j b (· ≤ a − b) = {σ ∈ T j b : d(σ jb , σ ) ≤ a − b} that is the closed ball of (T j b , d, σ jb ) with center σ jb and radius a − b. For any integer n ≥ 1, for any λ ∈ (0, ∞) and for any μ ∈ [0, ∞), we then set ¯ ρ , b) )) + λ Kn = λ (n ∧ Hg (B(

∑ 1{supH b, j >1/n} n ∧ Hg(T jb(· ≤ a − b) )

j∈Ib

+μ

∑ 1{sup H b, j >1/n} La−b ζ j ( j).

j∈Ib

j,b at level a − b. Since H is continHere La−b ζ j ( j) stands for the local time of H uous, the sum only contains a finite number of non-zero terms. Recall that since Hg (T j b (· ≤ a − b)) is a measurable function of H b, j , the branching property (2.20) implies that for any λ ∈ (0, ∞),

¯ ρ ,b) )−Lb κ˜ (n) (λ , μ ) −λ · n∧Hg(B( ζ a−b , Nγ(b) − a.e. Nγ(b) e−Kn Gb = e where ¯ ρ ,a−b))− μ La−b −λ · n∧Hg(B( (n) ζ . κ˜ a−b(λ , μ ) = Nγ 1{sup H>1/n} (1 − e (n) (λ , μ ) = κ˜ a−b (λ , μ ). Then note that Monotone convergence implies limn→∞ κ˜ a−b Nγ -a.e. ¯ ρ , a) ) + μ La . lim Kn = λ Hg (B( ζ

n→∞


129

The conditional dominated convergence theorem implies that for any λ ∈ (0, ∞), any μ ∈ [0, ∞) and any a > b > 0, we Nγ(b) -a.s. have ¯ ρ ,a))− μ La −λ Hg (B( ζ G Nγ(b) 1{Hg (B( ¯ ρ ,a)) 0, then Nγ (Hg (T ) = ∞) = 0. (Claim 2) Nγ (0 < Hg (T ) < ∞) = 0. We first prove (Claim 1). Let us suppose that Nγ (Hg (T ) < ∞) > 0. Then, there ¯ ρ , a0 ) < ∞ ; sup H > b0 ) > 0. Next, observe exists a0 > b0 > 0 such that Nγ (Hg (B( that the left member in (3.25) with a = a0 and b = b0 is strictly positive, which entails that κ˜ a0 −b0 (λ , 0) < ∞, for any λ ∈ (0, ∞), since we Nγ(b) -a.s. have Lbζ > 0. Therefore, we have ¯ ρ , a0 − b0 )) = ∞) ≤ κ˜ a0 −b0 (λ , 0) < ∞ , Nγ (Hg (B(

λ ∈ (0, ∞).

The scaling property easily entails that ¯ ρ , b)) = ∞) = b−1/(γ −1) Nγ (Hg (B( ¯ ρ , 1)) = ∞). Nγ (Hg (B( ¯ ρ , b)) = ∞) is obviously non-decreasing and finite at Since b → Nγ (Hg (B( ¯ ρ , b)) = ∞) = 0 for any b > 0. Consequently, b = a0 − b0, we get Nγ (Hg (B( Nγ (Hg (T ) = ∞) = 0, since T is bounded. This completes the proof of the first claim. We now prove (Claim 2). We argue by contradiction, so we suppose that Nγ (0 < Hg (T ) < ∞) > 0.

(3.27)

¯ ρ , a)) = ∞) = 0 for any a > 0. First, observe that (Claim 1) entails that Nγ (Hg (B( Since T is bounded, we get Nγ − a.e. Hg (T ) < ∞.

(3.28)

Then observe that for any b ∈ (0, ∞) the left member in (3.25) is strictly positive for any λ > 0. This entails κ˜ a−b (λ , 0) < ∞ for any a > b > 0, λ > 0, since we Nγ(b) -a.s. have Lbζ > 0, as already mentionned. More simply, we have proved

130

T. Duquesne

κ˜ a (λ , 0) < ∞ ,

a, λ ∈ (0, ∞).

Let μ > 0. Observe that 0 ≤ κ˜ a (λ , μ ) − κ˜ a (λ , 0) ≤ μ Nγ (Laζ ) = μ .

(3.29)

This implies that κ˜ a (λ , μ ) < ∞. Moreover (3.27) implies that κ˜ a (λ , μ ) > 0. Thus, we have proved that

κ˜ a (λ , μ ) ∈ (0, ∞) ,

a, λ > 0 , μ ≥ 0.

Since Nγ (Lbζ = 0) = Nγ (sup H > b), (3.25) easily entails ¯ ρ ,b) )−Lb κ˜ a−b (λ , μ ) −λ Hg(B( ζ = κ˜ b (λ , κ˜ a−b (λ , μ ) ). κ˜ a (λ , μ ) = Nγ 1 − e

(3.30)

We next prove that for any λ ∈ (0, ∞), there exists φ (λ ) ∈ (0, ∞), such that for any a > 0, we have a ¯ ¯ κ˜ a (λ , 0) = Nγ 1 − e−λ Hg(B(ρ ,a) ) = φ (λ ) Nγ e−λ Hg (B(ρ ,b) ) Lbζ db.

(3.31)

0

¯ Proof of (3.31). We first set U(λ , a) = Nγ e−λ Hg (B(ρ ,a) ) Laζ , for any a, λ ∈ (0, ∞). Observe that (3.25), (3.28) and (3.29) imply for any a > b > 0, and any μ ∈ (0, ∞) ¯ − μ La 1 Nγ(b) e−λ Hg (B(ρ ,a)) μ 1 − e ζ Gb =e

¯ ρ ,b))−Lb κ˜ a−b (λ ,0) 1 −λ Hg (B( ζ μ

≤ e−λ Hg(B(ρ ,b)) Lbζ ¯

−Lb (κ˜ (λ , μ )−κ˜ a−b (λ ,0) ) 1 − e ζ a−b

1 (κ˜ a−b (λ , μ ) − κ˜ a−b(λ , 0) ) μ

≤ e−λ Hg(B(ρ ,b)) Lbζ . ¯

Observe that the following limit is non-decreasing: limμ ↓0 ↑ Conditional monotone convergence entails

1 μ

− μ La 1 − e ζ = Laζ .

¯ ¯ Nγ(b) e−λ Hg(B(ρ ,a)) Laζ Gb ≤ e−λ Hg(B(ρ ,b)) Lbζ . We integrate this inequality with respect to Nγ(b) . Since {Laζ > 0} ⊂ {Lbζ > 0}, we easily get for any a > b > 0, and any λ ∈ (0, ∞), ¯ ¯ U(λ , a) = Nγ e−λ Hg(B(ρ ,a)) Laζ ≤ Nγ e−λ Hg(B(ρ ,b)) Lbζ = U(λ , b).

(3.32)


131

¯ ρ , b + h) ) − Hg (B( ¯ ρ , b) ) = 0, on the Next, let b, h ∈ (0, ∞) and note that Hg (B( / = Nγ (sup H > b), (3.25) and an elementary event {T (b) = 0}. / Since Nγ (T (b) = 0) inequality entail −Lb κ˜ (λ ,0) ¯ ¯ ¯ Nγ e−λ Hg (B(ρ ,b)) − e−λ Hg(B(ρ ,b+h)) = Nγ e−λ Hg(B(ρ ,b)) 1−e ζ h ≤ κ˜ h (λ , 0) (here we use the fact Nγ (Lbζ ) = 1 in the last inequality). This implies

κ˜ a (λ , 0) = κ˜ 1 (λ , 0) + n κ˜ 1 (λ , 0) n

n

na−1 n

Gn (b, λ )db + Rn(a, λ ),

0

(3.33)

where we have set ⎧ nb/n −L κ˜ 1/n (λ ,0) ⎨ ¯ Gn (b, λ ) = (κ˜ 1/n (λ , 0) )−1 Nγ e−λ Hg(B(ρ ,nb/n)) 1−e ζ , na/n −λ H (B( ˜ −L κ ( λ ,0) ⎩ ¯ {an}/n ρ ,na/n)) 1−e ζ g , Rn (a, λ ) = Nγ e where · stands for the integer-part function, where · = · + 1 and where {an} = na − na stands for the fractional part of na. ¯ ρ , h)) = Hg ({ρ }) = 0, since any Observe that Nγ -a.e. we have limh→0 Hg (B( Hausdorff measure is diffuse. Dominated convergence entails that limh→0 κ˜ h (λ , 0) = 0, for any λ ∈ (0, ∞). Consequently, limn κ˜ 1/n (λ , 0) = 0. Moreover, we get = κ˜ {an} (λ , 0) −−−→ 0. Rn (a, λ ) ≤ κ˜ {an} (λ , 0) Nγ Lna/n ζ n

n→∞

n

Next, observe that nb ¯ = U(λ , n ). Gn (b, λ ) ≤ Nγ e−λ Hg (B(ρ ,nb/n)) Lnb/n ζ By (3.32), we get Gn (b, λ ) ≤ U(λ , b). By (3.28), Hg is a finite measure. Then, ¯ ρ , b) ) is right continuous. Recall that we work with a right-continuous b → Hg (B( modification of b → Lbζ . Therefore, Fatou’s Lemma implies ¯ U(λ , b) = Nγ e−λ Hg (B(ρ ,b) ) Lbζ ≤ lim inf Gn (b, λ ), n

Thus, we get ¯ lim Gn (b, λ ) = U(λ , b) = Nγ e−λ Hg (B(ρ ,b) ) Lbζ . n

By dominated convergence, we get

lim n

0

na−1 n

Gn (b, λ )db =

a 0

¯ Nγ e−λ Hg (B(ρ ,b) ) Lbζ db.

132

T. Duquesne

This limit combined with (3.33) implies limn nκ˜ 1 (λ , 0) = φ (λ ) ∈ (0, ∞), which n implies (3.31). Next observe that (3.31) implies (∂ κ˜ a /∂ a)(λ , 0) = φ (λ )∂ (κã /∂ μ )(λ , 0) and by the scaling property (3.26) we easily get

φ (λ ) = c0 λ ,

with c0 := φ (1) ∈ (0, ∞) .

(3.34)

We next prove that for any b ∈ (0, ∞), ¯ ρ , b) ) = c0 m(B( ¯ ρ , b) ). Nγ − a.e. Hg (B(

(3.35)

Proof of (3.35). We first prove this result for b = 1. To that end, we set Dn (λ ) = 1−e−λ Hg(B(ρ ,1)) −Sn (λ ) ¯

where Sn (λ ) stands for Sn (λ ) =

∑

k/n −L κ˜ (λ ,0) ¯ . e−λ Hg (B(ρ ,k/n)) 1 − e ζ 1/n

1≤k 1. Next observe that [[ρ , σ [[⊂ Sk(T ). Consequently, we get m([[ρ , σ ]]) = 0. We thus have proved that on A, Hg and c0 m are finite Borel measures on T that agree on the set of all closed balls of T . This clearly implies (3.37). We have proved that (3.27) implies Hg = c0 m Nγ -a.e. We now argue as in the proof of Theorem 1.6 to get a contradiction thanks to the test stated in Proposition 1.9. This proves that (3.27) is wrong, which entails (Claim 2). As already mentioned, it completes the proof of Theorem 1.10. Acknowledgements This work benefited from the partial support of ANR-BLAN. We would like to thank an anonymous referee for helpful comments and for pointing out relevant references.

References 1. Abraham, R., Delmas, J.-F.: Fragmentation associated to Lévy processes using snake. Probab. Theor. Relat. Field. 141, 113–154 (2008) 2. Aldous, D.J.: The continuum random tree I. Ann. Probab. 19, 1–28 (1991) 3. Aldous, D.J.: The continuum random tree III. Ann. Probab. 21, 248–289 (1993) 4. Berlinkov, A.: Exact packing dimension in random recursive constructions. Probab. Theor. Relat. Field. 126, 477–496 (2003) 5. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) 6. Bingham, N.H.: Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4, 217–242 (1976) 7. Bingham, N.H., Doney, R.A.: Asymptotic properties of super-critical branching processes. I: The Galton-Watson process. Annals of Applied Probab. 6, 711–731 (1974) 8. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopaedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1987) 9. Dress, A., Moulton, V., Terhalle, W.: T-theory: An overview. Eur. J. Combin. 17, 161–175 (1996) 10. Duquesne, T.: A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31(2), 996–1027 (2003) 11. Duquesne, T.: The exact packing measure of Lévy trees. Preprint available on ArXiv (2010) 12. Duquesne, T., Le Gall, J.-F.: Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281 (2002) 13. Duquesne, T., Le Gall, J.-F.: Probabilistic and fractal aspects of Lévy trees. Probab. Theor. Rel. Field. 131(4), 553–603 (2005) 14. Duquesne, T., Le Gall, J.-F.: The Hausdorff measure of stable trees. Alea 1, 393–415 (2006) 15. Duquesne, T., Le Gall, J.-F.: On the re-rooting invariance property of Lévy trees. Elect. Comm. Probab. 14, 317–326 (2009) 16. Duquesne, T., Winkel, M.: Growth of Lévy trees. Prob. Theor. Relat. Field. 139(3–4), 313–371 (2007) 17. Edgar, G.: Centered densities and fractal measures. New York J. Math. 13, 33–87 (2007) 18. Evans, S.: Probability and Real Trees. Saint-Flour Lectures Notes XXXV. Springer, Berlin (2005) 19. Evans, S., Pitman, J., Winter, A.: Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theor. Relat. Field. 134, 81–126 (2006)

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20. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971) 21. Goldschmidt, C., Haas, B.: Behavior near extinction time in self-similar fragmentation I: the stable case. Ann. IHP, 46(2), pp. 338–368 (2010) 22. Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Birkhäuser, Boston (1999) 23. Haas, B., Miermont, G.: The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electr. J. Probab. 9, 57–97 (2004) 24. Haase, H.: The packing theorem and packing measure. Math. Nachr. 146, 77–84 (1990) 25. Helland, I.S.: Continuity of a class of random time transformations. Stoch. Process. Appl. 7, 79–99 (1978) 26. Jirina, M.: Stochastic branching processes with continous state-space. Czech. Math. J. 8, 292–313 (1958) 27. Joyce, H.: Concerning the problem of subsets of finite positive packing measure. J. Lond. Math. Soc. 56, 557–566 (1997) 28. Joyce, H.: A space on which diameter-type packing measure is not Borel regular. Proc. Amer. Math. Soc. 127, 985–991 (1999) 29. Joyce, H., Preiss, D.: On the existence of subsets of positive finite packing measure. Mathematika 42, 14–24 (1995) 30. Lamperti, J.: Continuous-state branching processes. Bull. Amer. Math. Soc. 73, 382–386 (1967) 31. Larman, D.G.: A new theory of dimension. Proc. Lond. Math. Soc. 17, 178–192 (1967) 32. Le Gall, J.-F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics. ETH, Zürich (1999) 33. Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26(1), 213–252 (1998) 34. Le Gall, J.-F., Perkins, E., Taylor, S.: The packing measure of the support of super-Brownian motion. Stoch. Process. Appl. 59, 1–20 (1995) 35. Miermont, G.: Self-similar fragmentations derived from the stable tree I: Splitting at heights. Probab. Theor. Relat. Field. 127(3), 423–454 (2003) 36. Miermont, G.: Self-similar fragmentations derived from the stable tree II: Splitting at nodes. Probab. Theor. Relat. Field. 131(3), 341–375 (2005) 37. Rogers, C., Taylor, S.: Functions continuous and singular with respect to Hausdorff measures. Mathematika 8, 1–31 (1961) 38. Silverstein, M.: A new approach to local times. J. Math. Mech. 17, 1023–1054 (1968) 39. Taylor, S., Tricot, C.: Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288, 679–699 (1985) 40. Weill, M.: Regenerative real trees. Ann. Probab. 35(6), 2091–2121 (2007)

Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing Oleg Reichmann and Christoph Schwab

Abstract We review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogorov equations arising, among others, in financial engineering when Lévy and Feller or additive processes are used to model the dynamics of the risky assets. In particular, the Dirichlet and free boundary problems connected to barrier and American style contracts are specified and solution algorithms based on wavelet representations of the Feller processes’ Dirichlet forms are presented. Feller processes with generators that give rise to Sobolev spaces of variable differentiation order (corresponding to a state-dependent jump intensity) are considered. A copula construction for the systematic construction of parametric multivariate Feller-Lévy processes from univariate ones is presented and the domains of the generators of the resulting multivariate Feller-Lévy processes is identified. New multiresolution norm equivalences in such Sobolev spaces allow for wavelet compression of the matrix representations of the Dirichlet forms. Implementational aspects, in particular the regularization of the process’ Dirichlet form and the singularity-free, fast numerical evaluation of moments of the Dirichlet form with respect to piecewise linear, continuous biorthogonal wavelet bases are addressed. Monte Carlo path simulation techniques for such processes by FFT and symbol localization are outlined. Numerical experiments illustrate multilevel preconditioning of the moment matrices for several exotic contracts as well as for Feller-Lévy processes with variable order jump intensities. Model sensitivity of Lévy models embedded into Feller classes is studied numerically for several types of plain vanilla, barrier and exotic contracts. AMS Subject Classification 2000: Primary: 45K05, 60J75, 65N30 Secondary: 45M60

O. Reichmann and C. Schwab () ETH Zürich, Seminar für Angewandte Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland e-mail: [email protected]; [email protected]


137

138

O. Reichmann and C. Schwab

Keywords Dirichlet forms · Feller processes · Kolmogorov equations · Lévy processes · Matrix compression · Option pricing · Sato processes · Wavelet discretization

1 Introduction We consider a certain type of multidimensional normal Markov processes, so called Feller processes. This class of Feller processes includes as special cases Lévy processes, many local volatility and, in particular, the so-called affine models in finance as special cases. Due to their nonstationarity Feller processes can exhibit qualitative behaviour that is substantially different from that of Lévy processes such as state-space dependent jump activity. The nonstationarity of Feller processes also has substantial repercussions on their computational and analytical treatment: whereas for Lévy and the closely related affine models, Fast Fourier Transformation (FFT) algorithms from [24] (in modern, hardware optimized implementations, e.g. [39]) form the basis for fast and powerful option pricing algorithms, the nonstationarity of Feller processes implies that FFT based numerical methods are, in general, not applicable in the numerical solution of their Kolmogorov equations (with the notable exception of, for example, affine processes proposed e.g. in [34] and, for Feller processes, in connection with approaches based on “freezing” their characteristic triplett [14]). From an analytical point of view, Feller processes are rather well understood. This is due to the fact that generators of Feller processes are pseudodifferential operators with symbols that admit a Lévy-Khintchine representation (e.g. [12, 25, 51, 52] and the references there). Contrary to Lévy processes or diffusions with local volatility, domains of the infinitesimal generators for semigroups induced by Feller processes are, generally, generally anisotropic, often variable order Sobolev spaces. Accordingly, the use of standard discretization schemes (based on Finite Differences or Finite Elements) for numerical solution of the Kolmogorov equations associated to such Feller processes is not straightforward; the same applies to the numerical analysis of these discretization schemes, i.e. the mathematical analysis of stability, consistency and convergence of these schemes. One central theme of these notes is therefore to describe recent progress in the design and the numerical analysis of discretization schemes which allow a unified numerical treatment of the Kolmogorov equations of Feller (and more general) processes. These schemes are based on variational, multiresolution schemes which use spline-wavelet bases of the domains of the processes’ infinitesimal generators. Feller processes arise as natural generalizations of Lévy processes [20] and are also useful for modeling bounded processes [7]. Feller processes appear as solutions of a large class of Lévy SDEs as shown in the recent work [83, 85]. Therefore, the high dimensional problem of pricing basket options under a Lévy market model can be reduced to a low dimensional problem driven by a Feller process. Besides, due to mimicking results by [41] for continuous semimartingales and a novel result by [6] for discontinuous semimartingales, pricing of European options under general

Numerical Analysis of Feller Processes

139

non-Markovian processes can be reduced to a Markovian setting with processes that have deterministic, but time and state-space dependent coefficients. However, we will mainly focus on the case of time-independent coefficients in the following. The theoretical literature on Feller processes is quite extensive. We refer to the monographs by N. Jacob [51–53] and [54], as well as references therein for an overview of properties of the generators and the corresponding semigroups. The Martingale Problem, i.e., the problem of existence of a Feller process with a given generator, has been treated by [46, 58, 90]. Path properties have been discussed in [79–81]. On the other hand, numerical methods capable of handling general Feller processes have received little attention until now. A technique for the approximation of sample paths of Feller processes is given in [14]. A fast calibration algorithm has been proposed for a special case in [19, 26]. Fourier methods for option pricing can only be efficiently used for regular affine processes, i.e., Feller processes with coefficients that depend affinely on the state variables [34, 35]. Finite Element based pricing methods for one dimensional European pricing problems under general Feller processes are proposed by [84], while [73] considers multidimensional pricing problems of European and American type under Lévy processes using FEM. Multidimensional Feller processes can be constructed using Lévy copulas for the construction of the jump measures. We consider European and American type contracts, as well as the calculation of sensitivities and discuss well-posedness of the corresponding pricing Partial Integro-Differential equations (PIDEs) and inequalities in the multidimensional Feller setting. In addition, we discuss wavelet based discretization schemes, that allow for efficient preconditioning. Using wavelets, the arising densely populated matrices can be compressed leading to computational schemes with essentially Black- Scholes complexity. Efficient numerical quadrature rules for the evaluation of the wavelet coefficients of the jump measures of this survey are presented. The outline is as follows. First, we give an overview of Feller processes and state sufficient conditions on the characteristic triplet for the existence of a corresponding Feller process (Section 2). Next, we define the domains of the generators, which are Sobolev spaces of variable order, we employ pseudodifferential operator (PDO) theory to characterize the spaces (Section 3). Then we describe parametric constructions of multidimensional Feller processes using Lévy copulas and state sufficient conditions on the marginals and the copula function to obtain admissible market models. We show that the class of admissible market models contains many typical examples of Lévy market models and the extensions to Feller market models (Section 4). We discuss European and American pricing problems, as well as the calculation of sensitivities with respect to model parameters and solution arguments. The G˚arding inequality and the sector condition is proven for the arising bilinear forms, which yields well-posedness of the corresponding pricing problems (Section 5). In Section 6 we introduce a tensor product wavelet basis and discuss norm equivalences on the variable order Sobolev spaces. The discretization of the arising non-local operators using this basis is presented. In Section 7 we address quadrature rules for the weakly singular integrals in the Galerkin discretization and

140


briefly survey on Monte Carlo and Fourier methods in Section 8. We conclude with uni- and bivariate numerical examples from the pricing of derivative contracts. Throughout, we shall write C D to denote that C can be bounded by a constant multiple of D with a constant that is independent of parameters which C and D may depend on. Then C D is defined as D C and C D is defined as C D and D C.

2 Markov Processes The following section introduces the stochastic processes that will be considered in this work. It turns out that the processes can be characterized via the symbol of their generator. Semimartingales are a well-investigated class of stochastic processes that is sufficiently rich to include most of the stochastic processes commonly employed in financial modelling while still being closed under various operations such as conditional expectations, stopping etc. Semimartingales can be well understood via their (generally stochastic) semimartingale characteristic, we refer to the standard reference [55] for details. Here, we restrict ourselves to a class of processes with deterministic, but generally time- and state-space dependent characteristic triplets including Lévy processes, affine processes and many local volatility models. The time-homogeneous case will be analyzed in the first part of this section, while timeinhomogeneity will be briefly discussed in the second part.

2.1 Time-Homogeneous Processes We consider a Markov process X and the corresponding family of operators (Ts,t ) for 0 ≤ s ≤ t < ∞ given by (Ts,t ( f ))(x) = E[ f (Xt )|X(s) = x], for each f ∈ Bb (Rd ), x ∈ Rd , where Bb (Rd ) denotes the space of bounded Borel measurable functions on Rd . For Markov processes, whose semigroups satisfy Ts,t (Bb (Rd )) ⊂ Bb (Rd ), we recall the following properties: (1) (2) (3) (4) (5) (6)

Ts,t is a linear operator on Bb (Rd ) for each 0 ≤ s ≤ t < ∞. Ts,s = I for each s ≥ 0. Tr,s Ts,t = Tr,t whenever 0 ≤ r ≤ s ≤ t < ∞. f ≥ 0 implies Ts,t f ≥ 0 for all 0 ≤ s ≤ t < ∞, f ∈ Bb (Rd ). Ts,t ≤ 1 for each 0 ≤ s ≤ t < ∞, i.e. Ts,t is a contraction. Ts,t (1) = 1 for all t ≥ 0.

(2.1)


141

If we restrict ourselves to time-homogeneous Markov processes satisfying (2.1), we obtain directly from the above properties that the family of operators Tt := T0,t form a positivity preserving contraction semigroup. The infinitesimal generator A with domain D(A ) of such a process X with semigroup (Tt )t≥0 is defined by the strong pointwise limit 1 (2.2) A u := lim (Tt u − u) + t→0 t for all functions u ∈ D(A ) ⊂ Bb (Rd ) for which the limit (2.2) exists w.r. to the supnorm. We call (A , D(A )) generator of X. Generators of normal Markov processes admit the positive maximum principle, i.e., if

u ∈ D(A )

sup u(x) = u(x0 ) > 0,

and

then (A u)(x0 ) ≤ 0.

(2.3)

x∈Rd

Furthermore, they admit a pseudodifferential representation (e.g. [12, 25, 51, 52]): Theorem 2.1. Let (A , D(A )) be an operator with C0∞ (Rd ) ⊂ D(A ) and A (C0∞ (Rd )) ⊂ C(R). Then A |C∞ (Rd ) is a pseudodifferential operator, 0

(A u)(x) = −a(x, D)u(x) = −(2π )−1/2

Rd

a(x, ξ )u( ˆ ξ )eix·ξ d ξ

(2.4)

for u ∈ C0∞ (Rd ). With a symbol a(x, ξ ) : Rd × Rd → C which is locally bounded in (x, ξ ), a(·, ξ ) is measurable for every ξ and a(x, ·) is a negative definite function for every x, which admits the Lévy-Khintchine representation 1 a(x, ξ ) = c(x) − iγ (x) · ξ + ξ · Q(x)ξ 2 iy · ξ iy·ξ + 1−e + N(x, dy). 1 + y2 0 =y∈Rd

(2.5)

Here, for y ∈ Rd , y2 = y y and the function Rd x →

y =0

(1 ∧ y2 )N(x, dy)

(2.6)

is continuous and bounded. The parameters c(x), γ (x), Q(x), N(x, dy) in (2.5) are called characteristics of the Markov process X. In the following we set c(x) = 0 for notational convenience and restrict ourselves to a certain kind of normal Markov processes, so called Feller processes ([2, Theorem 3.1.8] states (2.1) for Feller process, see also [74, p. 83]). These can be defined by the semigroup (Tt )t≥0 generated by the corresponding process X. A semigroup (Tt )t≥0 is called Feller if it satisfies

142


(i) Tt maps C∞ (Rd ), the continuous functions on Rd vanishing at infinity, into itself: Tt : C∞ (Rd ) → C∞ (Rd )

boundedly

(ii) Tt is strongly continuous, i.e., limt→0+ u − Tt uL∞ (Rd ) = 0 for all u ∈ C∞ (Rd ). Spatially homogeneous Feller processes are Lévy-processes (e.g.[9,76]). Their characteristics, the Lévy characteristics, do not depend on x explicitly. Example 2.2 A standard Brownian motion has the characteristics (0, 1, 0). An R-valued Lévy process has characteristics (γ , Q, N(dy)), for real numbers γ , Q ≥ 0 and a jump measure N with 0 =y∈R min(1, y2 )N(dy) < ∞. A recent result by Schnurr [83,85] additionally motivates the consideration of Feller processes. He proved that strong solutions of a large class of Lévy SDEs are in fact Feller processes, i.e. Theorem 2.3. Let Z be an n-dimensional Lévy process and Φ : Rd → Rd×n be bounded and globally Lipschitz. Then the solution of Xt = x +

t 0

Φ (Xs− ) dZs ,

x ∈ Rd is a Feller process with C0∞ (Rd ) ⊂ D(A ). Proof. The proof is given in [85] Theorem 2.46, Theorem 2.49 and Theorem 2.50.

Remark 2.4. The boundedness of Φ can be replaced by certain assumptions on the tail behaviour of the process Z. This implies that pricing of a large class of basket options in a Lévy market model can be reduced to pricing under a low dimensional Feller processes. It is also interesting to ask which symbols correspond to PDOs that are generators of Feller processes. This martingale problem is discussed in the following theorem due to [82]. Theorem 2.5. Let a : Rd × Rd → C be a negative definite function, i.e., a measurable and locally bounded function that admits a Lévy-Khinchine representation for all x ∈ Rd ; note that this implies continuity of the symbol in ξ for all x ∈ Rd . If (a) supx∈Rd |a(x, ξ )| ≤ κ (1 + |ξ |2 ) for all ξ ∈ Rd , (b) ξ → a(x, ξ ) is uniformly continuous at ξ = 0, (c) x → a(x, ξ ) is continuous for all ξ ∈ Rd , then (−a(x, D),C0∞ (Rd )) extends to a Feller generator. Remark 2.6. In the Lévy case existence of a Lévy process can be proven for any Lévy symbol. This does not hold for Feller processes. For (financial) applications it is more convenient to consider the characteristic triplet instead of the symbol. We therefore make the following assumption on the characteristic triplet in the remainder.


143

Remark 2.7. Note that Theorem 2.5 does not imply the well-posedness of the martingale problem, i.e., the uniqueness of the process. We will show wellposedness of the pricing equations in Section 5. For the existence of a solution of the martingale problem it is sufficient to require (a) from Theorem 2.5 and a(x, 0) = 0 for all x ∈ Rd , cf. [45, Theorem 3.15], (b) and (c) are required to obtain a Feller process. Assumption 2.8 The characteristic triplet (γ (x), Q(x), N(x, dy)) of a Feller process in Rd satisfies the following conditions: (I) (γ (x), Q(x), N(x, dy)) is a Lévy triplet for all fixed x ∈ Rd . (II) The mapping x → B∩Rd \{0} (1 ∧ |y|2 )N(x, dy) is continuous for all B ∈ B(Rd ). (III) There exists a Lévy measure N(y) s.t. B∩Rd \{0}

(1 ∧ |y|2 )N(x, dy) ≤

B∩Rd \{0}

(1 ∧ |y|2 )N(dy) < ∞

for all x ∈ Rd , B ∈ B(Rd ). (IV) The functions x → γ (x) and x → Q(x) are continuous and bounded. We would like to conclude that there exists a Feller process whose generator is a PDO for symbols that satisfies Assumption 2.8. Therefore, it suffices to validate the prerequisites of Theorem 2.5. Lemma 2.9. Let (γ (x), Q(x), N(x, dy)) be the characteristic triplet of a process X taking values in Rd that satisfies Assumption 2.8. Then (−a(x, D),Cc∞ ) extends to a Feller generator, where a(x, ξ ) is given by 1 a(x, ξ ) = −iγ (x) · ξ + ξ · Q(x)ξ 2 +

Rd \{0}

1 − eiy·ξ +

iy · ξ

1 + |y|2

(2.7)

N(x, dy).

Proof. Condition (I) implies that the corresponding Feller symbol is negative definite. Conditions (III) and (IV) imply (a), Conditions (II) and (III) imply (b), and (c) follows from (IV) and (II).

Remark 2.10. Note that real price market models do not fit into our modeling framework due to Assumption (a) in Theoren 2.5, but the numerical methods and their numerical analysis presented in the following can in many cases be straightforwardly extended to this kind of models. As illustrative example we consider the CEV market model (e.g. [32]) throughout this survey and explain the necessary extensions. The CEV model is given by the following SDE: ρ

dSt = rSt dt + σ St dWt ,

S0 = s ≥ 0,

(2.8)

144


where ρ ∈ (0, 1), σ > 0 and r ≥ 0. Existence of a solution of (2.9) follows from the Skorohod existence theorem [47, Theorem IV.2.2], while pathwise uniqueness can be obtained for ρ ≥ 0.5 from the Yamada conditions (e.g. [47, Theorem IV.3.2]). Note that the case ρ = 0.5 leads to equations similar to the Heston model and CIR model. In order to apply pseudodifferential operator theory we will need stronger assumptions on the characteristic triplets of the considered processes. We will state the assumptions needed at the end of Section 4. We will require in particular smoothness of the characteristic triplet in the state variable x. Numerical experiments indicate that these assumptions can be weakened (Chapter 9).

2.2 Time-Inhomogeneous Processes In this section we would like to drop the assumption of time-homogeneity of the processes considered and extend the framework developed above to a time-dependent setting. Using the notation from the last section, we consider a normal Markov process X with the corresponding family of operators Ts,t . The family of generators of such a process is given by As u := lim

h→0+

1 Ts−h,s u − u h

(2.9)

for all functions u ∈ D(As ) ⊂ Bb (Rd ), such that the limit exists in the strong pointwise sense. In analogy to Theorem 2.1 we obtain the following result: Theorem 2.11. Let (As , D(As ))s∈R+ be a family of operators with C0∞ (Rd ) ⊂ D(As ) and As (C0∞ (Rd )) ⊂ C(R). Then As |C∞ (Rd ) is a pseudodifferential operator 0 for all s ∈ R+ given by (As u)(x) = −a(s, x, D)u(x) = −(2π )−1/2

Rd

a(s, x, ξ )u( ˆ ξ )eix·ξ d ξ

for u ∈ C0∞ (Rd ). With a symbol a(s, x, ξ ) : R+ × Rd × Rd → C which is locally bounded in (x, ξ ), a(s, ·, ξ ) is measurable for every ξ , s and a(s, x, ·) is a negative definite function for every (s, x), which admits the Lévy-Khintchine representation 1 a(s, x, ξ ) = c(s, x) − iγ (s, x) · ξ + ξ · Q(s, x)ξ 2 iy · ξ iy·ξ + 1−e + N(s, x, dy). 1 + y2 0 =y∈Rd The natural question arises if we can construct a Markov process with corresponding generator for a given symbol (Martingale Problem). A general result under mild regularity assumptions on the symbol has been given by [13].


145

Theorem 2.12. Let a : R+ × Rd × Rd → C be a negative definite function that satisfies the following conditions for a constant m ∈ R: (1) a(·, x, ξ ) is a continuous function for all x, ξ ∈ Rd , + d (2) a(s, x, 0) = 0 holds

for all (s, x) ∈ R × R ,

β

(3) Dx Dξα a(s, x, ξ ) ≤ cα ,β ,J (1 + |ξ |2 )(m−|α |∧2)/2 holds for all s ∈ J ⊂ R+ ,

x, ξ ∈ Rd , (4) a is elliptic, i.e., on any compact set K it holds uniformly in s that: there exists R ∈ R+ , c > 0, such that ∀x ∈ Rd , |ξ | ≥ R : ℜ(a(s, x, ξ )) ≥ c(1 + |ξ |2 )m/2 .

Then a Markov process whose family of generators are pseudodifferential operators with symbol a(s, x, ξ ) can be constructed.

Proof. The proof follows from [13, Theorem 4.2, Corollary 4.3].

Theorem 2.12 can be formulated in a more general setting, replacing |ξ | in Condition (3) and (4) by any element from a certain class of negative definite functions, cf. [13, Definition 1.2]. 2

Remark 2.13. In the following we will consider the time-homogeneous case discussed in Section 2.1, but most results can be extended to the inhomogeneous setting.

3 Function Spaces It turns out that the domains of the bilinear forms considered in the following sections are generally anisotropic, often variable order Sobolov spaces. Therefore these spaces are defined in this section. For our analysis we will need certain Sobolev-type spaces. Therefore we start with the definition of fractional order isotropic spaces. We define for a positive noninteger s ≥ 0 and u ∈ S ∗ (Rd )

u2H s (Rd ) :=

Rd

(1 + |ξ |2 )s |u( ˆ ξ )|2 dξ ,

(3.1)

where uˆ is the Fourier transform of u. Similarly, we can define anisotropic Sobolev spaces H s (Rd ) with norm u2H s (Rd ) :=

d

ˆ ξ )|2 dξ , ∑ (1 + ξ j2)s j |u( Rd

(3.2)

j=1

for any multiindex s ≥ 0. The consideration of certain symbol classes will be useful for the definition of the variable order Sobolev spaces. We set ξ := (1 + |ξ |2 )1/2 for notational convenience.

146


Definition 3.1. Let 0 ≤ δ < ρ ≤ 1 and let m(x) ∈ C∞ (Rd ) be a real-valued function all of which derivatives are bounded on Rd . The symbol a(x, ξ ) belongs to the class m(x) Sρ ,δ of symbols of variable order m(x) if a(x, ξ ) ∈ C∞ (Rd × Rd ) and m(x) = s +

m(x) with m ∈ S (Rd ) a tempered function, and if, for every α , β ∈ Nd0 there is a constant cα ,β such that |Dβx Dξα a(x, ξ )| ≤ cα ,β ξ m(x)−ρ |α |+δ |β | .

∀x, ξ ∈ Rd :

(3.3)

m(x)

The variable order pseudodifferential operators A(x, D) ∈ Ψρ ,δ correspond to symm(x)

bols a(x, ξ ) ∈ Sρ ,δ by A(x, D)u(x) :=

1 2π

Rd

Rd

ei(x−y)·ξ a(x, ξ )u(y)dyd ξ ,

u ∈ C0∞ (Rd ).

(3.4)

We are now able to define an isotropic Sobolev space of variable order using the variable order Riesz potential Λ m(x) with symbol a(x, ξ ) = ξ m(x) . Clearly a(x, ξ ) m(x) is an element of S1,δ for δ ∈ (0, 1). The norm on H m(x) is given as 2 u2H m(x) := Λ m(x) u 2 + u2L2 . L

Note that for a(x, ξ ) = 1, we obtain the usual L2 norm, while for a(x, ξ ) = (1 + |ξ |s ) we obtain the norm given in (3.1). The second result follows using Plancherel’s theorem. Now we turn to the definition of anisotropic variable order Sobolev spaces. In analogy to Definition 3.1 we start with the definition of an appropriate symbol class. Definition 3.2. Let m(x) = s+ m(x), m(x) : Rd → Rd with each component of m(x) being a tempered function and s ∈ Rd≥0 , 0 ≤ δ < ρ ≤ 1. We define the symbol class Sρ ,δ as the set of all a(x, ξ ) ∈ C∞ (Rd × Rd ) such that for all multiindices α , β ∈ Nd0 there exists a constant Cα ,β > 0 with m(x)

∀x, ξ ∈ Rd :

β α

Dx Dξ a(x, ξ ) ≤ Cα ,β

d

∑ (1 + ξi2)(mi (x)−ραi+δ |β |)/2 .

(3.5)

i=1

We are now able to define an anisotropic Sobolev space of variable order using the variable order Riesz potential Λ m(x) with symbol a(x, ξ ) = ξ m(x) := ∑ni=1 (1 + 1 m(x) ξi2 ) 2 mi (x) . Clearly, a(x, ξ ) is an element of S1,δ for δ ∈ (0, 1). The norm on H m(x) is given by 2 u2H m(x) := Λ m(x) u 2 + u2L2 . L


147

There is an alternative representation of the above space, when m(x) is of the following form m(x) = (m1 (x1 ), . . . , md (xd )) which will be very useful for the proof of norm equivalences. This plays a crucial role in wavelet discretization theory. We m (x ) consider the anisotropic Sobolev spaces Hi i i of variable order mi (xi ) in direction xi , equipped with the following norms: m (x ) 2 u2 m(x) := Λi i i u 2 + u2L2 , Hi

L

m (x )

where Λi i i is a pseudo-differential operator with symbol (1 + |ξi |)mi (xi ) . It then follows by the elementary inequality

d 2

d 2 d

2 C1 ∑ ai ≤ ∑ ai ≤ C2 ∑ ai ,

i=1

i=1 i=1 with ai > 0 and C1 ,C2 only dependent on d, that u2H m(x) ∼

d

∑ u2H m j (x j ) ,

j=1

j

and therefore H m(x) =

d

m j (x j )

Hj

.

j=1

On the bounded set D = (a, b) = ∏di=1 (ai , bi ) ⊂ Rd we define for a variable order m(x), a ≤ x ≤ b the space

m(x) (D) := u|D

u ∈ H m(x) (Rd ), H

u|Rd \D = 0 .

This space coincides with the closure of C0∞ (D) (the space of smooth functions with support compactly contained in D) with respect to the norm uH m(x) (Rd ) , uHm(x) (D) :=

(3.6)

where u is the zero extension of u to all of Rd . Remark 3.3. In the BS case we will obtain H 1 (Rd ) as the domain of the generator and H01 (D) in the localized case. In the Lévy case we obtain anisotropic Sobolev s (D) in the localized case for Q = 0. For Q > 0 spaces as in (3.2) and the spaces H the domains are equal to those in the BS case, cf. [73, Theorem 4.8].

148


4 Multivariate Model Setting 4.1 Copula Functions Unlike multivariate Lévy processes, not all multivariate Feller processes can be constructed in terms of univariate Feller processes using a homogeneous copula construction as in the case of Lévy processes in Rd , cf. [56, Theorem 3.6]. However, parametric constructions of multidimensional Feller processes from the univariate margins of certain Feller processes and certain Lévy copulas are still possible, provided the univariate Feller processes and the copulas meet certain restrictions. The restrictions stem from the fact that smoothness conditions on the characteristic triplet appear to be required in order to prove existence (and uniqueness) of a corresponding Feller process, cf. [82]. Therefore it would be sufficient for the parametric construction of d-dimensional Feller processes to prove that a symbol satisfies Assumption 2.8. We will only consider the construction of a d-dimensional jump measure, as the Gaussian part is standard. Theorem 4.1. Let F denote a d-dimensional Lévy copula for which the derivative d ∂1 . . . ∂d F : R → R exists, is continuous and satisfies the following estimate d

|∂ n F(u)| C|n| |n|! min{|u1 | , . . . , |ud |} ∏ |ui |−ni ∀u ∈ (R\{0})d n ∈ Nd . (4.1) i=1

Further let Ui (x, y), i = 1, . . . , d denote the tail integrals of real valued Feller processes that satisfy Assumption 2.8 such that ki (x, y)dy = Ni (x, dy) and Ui (x, y) =

∞ y k(x, y) dy y k(x, y) dy − −∞

y≥0 y 0 and i = 1, . . . , d. Then there exists an Rd -valued Feller process X whose components have tail integrals U1 , . . . ,Ud and whose marginal tail integrals satisfy U I ((xi )i∈I , (yi )i∈I ) = F I ((U(xi , yi ))i∈I ) for any non-empty I ⊂ {1, . . . , d}, any (yi )i∈I ∈ (R\{0})|I| and any (xi )i∈I ∈ R|I| . We denote by U I ((xi )i∈I , (yi )i∈I ) the tail integral of the process X I = (X i )i∈I . The jump measure is uniquely determined by F and Ui , i = 1, . . . , d.


149

Proof. The proof follows [56]. As noted there, the argument is not restricted to Lévy models but can be extended to more general processes. Since F is d-increasing and continuous, we can conclude that there exists a d unique measure μ on R \{∞, . . . , ∞} such that VF ((a, b]) = μ ((a, b]) for any a, b with a ≤ b. For the univariate tail integrals U(x, y), we define U

−1

(x, u) =

inf{y > 0 : u ≥ U(x, y)},

u ≥ 0,

inf{y < 0 : u ≥ U(x, y)} ∧ 0, u < 0.

Let N = f (μ ) be the image of μ under f : (x, u1 , . . . , ud ) → (U1−1 (x1 , u1 ), . . . ,Ud−1 (xd , ud )) and let N be the restriction of N to Rd × Rd \{0}. We need to prove that N is a Lévy measure for all x and that the marginal tail integrals UNI satisfy UNI ((x)i∈I , (yi )i∈I ) = F I ((Ui (xi , yi ))i∈I ). This will imply (I). Furthermore, we must prove continuity of the Lévy kernel in x (II) as well as boundedness in the sense of (III). The first part follows analogously to [56]. We assume for ease of notation that yi > 0, i ∈ I. Then UNI ((xi )i∈I , (yi )i∈I ) = N (xi )i∈I , {ξ ∈ Rd \{0} : ξi ∈ (yi , ∞), i ∈ I} d = μ {u ∈ R : Ui−1 (xi , ui ) ∈ (yi , ∞), i ∈ I} d = μ {u ∈ R : 0 < ui < Ui (xi , yi ), i ∈ I} d = μ {u ∈ R : 0 < ui ≤ Ui (xi , yi ), i ∈ I} = F I ((Ui (xi , yi ))i∈I ) . This proves in particular that the one-dimensional marginal tail integrals of N equal U1 , . . . ,Ud . Since the margins of N(x, y) are Lévy measures on R\{0} for all x ∈ Rd we obtain for every x ∈ Rd :

y∈Rd

|y|2 ∧ 1 N(x, dy) ≤

=

d

∑

2 yi ∧ 1 N(x, dy)

d

∑ y∈Rd

i=1

i=1 yi ∈R

2 yi ∧ 1 Ni (xi , dyi ) < ∞.

Hence, for x ∈ Rd , N(x, ·) is a Lévy measure on Rd . For the second part of the proof we use Remark 2.7 in [73] which gives us: k(x, y1 , . . . , yd ) = ∂1 . . . ∂d F|ξ1 =U1 (x1 ,y1 ),...,ξd =Ud (xd ,yd ) k1 (x1 , y1 ) . . . kd (xd , yd ).

(4.4)

150


Using the properties of F and the margins we can conclude that k(x, y1 , . . . , yd ) is continuous in x for all (y1 , . . . , yd ) ∈ (R\{0})d . It remains to prove (III). Due to (4.1) we have the following estimate with g := ∂1 . . . ∂d F: k(x, y1 , . . . , yd ) = g(U1 (x1 , y1 ), . . . ,Ud (xd , yd ))k1 (x1 , y1 ) . . . kd (xd , yd ) d

d

i=1

i=1

≤ C min{|U1 (x1 , y1 )| , . . ., |Ud (xd , yd )|} ∏ |Ui (xi , yi )|−1 ∏ ki (xi , yi )

1

≤ C min{ U 1 (y1 ) , . . . , U d (yd ) } ∏ C ∨ , |yi | i=1

(4.2)

d

(4.5)

where U i denotes the tail integrals of N i , i = 1, . . . , d. Using the properties of the Ni (dy) for i = 1, . . . , d we can conclude that (4.5) is a Lévy measure and therefore (IV) is valid for k(x, y). Uniqueness of the jump measure follows from the fact that it is uniquely determined by its marginal tail integrals (cf. [56, Lemma 3.5]).

We can prove the following decay property of the jump density constructed according to the above theorem. Lemma 4.2. Let k(x, y) be constructed according to Theorem 4.1. Besides we require the following estimate on the derivatives of ki (x, y): there exists C > 0 s.t. ∀x ∈ Rd , y ∈ Rd \{0} |∂xn ki (x, y)| ≤ Cn+1 n! |y|−mi (x)−δ n−1 ,

n

∂ ki (x, y) ≤ Cn+1 n! |y|−mi (x)−n−1 , y

(4.6) (4.7)

for some δ ∈ (0, 1) and maxi=1,...,d supxi ∈R mi (xi ) = m < 2 as well as mini=1,...,d infxi ∈R mi (xi ) = m > 0. Then it holds d

m n

∂ ∂ k(x, y) ≤ C|n|+1 |m|! |n|! y−m ∏ |yi |−ni −δ mi −1 , x

∞

y

∀yi = 0,

i=1

for multiindices n, m ∈ Nd0 Proof. Using the formula of Faà di Bruno [75] it can be shown

n

∂x (∂1 . . . ∂d F(U(x, y)))

i

n

∂xi Ui (x, y) mn

n! ∂xi Ui (x, y) m1 m

= ∑ (∂ ∂1 . . . ∂d F)(U(x, y)) ...

m1 ! . . . mn ! xi 1! n! d

n!m! α

α α m |zi |−α m1 −δ m1 . . . |yi |−α mn −δ nmn y− ≤ ∑ C1n+1 ∞ ∏ y j |yi | m1 ! . . . mn ! j d

α α −δ n ≤ C2n+1 n! y− ∏ y j , ∞ |yi | j=1


151

where we sum over all multiindices (m1 , . . . , mn ), m = ∑i mi with n = ∑ni=1 imi . An analogous calculation leads to d

n

∂ (∂1 . . . ∂d F(U(x, y))) ≤ Cn+1 n! y−α |yi |−n ∏ y j α . yi

∞

2

j=1

Using the Leibniz rule we obtain

n

∂x k(x, y) = ∂xn (∂1 . . . ∂d F(U(x, y))k1 (x1 , y1 ) . . .kd (xd , yd ))

i i

n

d n!

j n− j = ∑ ∂xi (∂1 . . . ∂d F(U(x, y)))∂xi ki (xi , yi ) ∏ km (xm , ym )

j=1 j!(n − j)!

m=1,m =d n

α − jδ ≤ C3n+1 n! ∑ y− ∞ |yi | j=1

α ∏ y j |yi |−α −1+δ (−n+ j)

d

d

j=1

∏

m=1,m =i

|ym |−α −1

d

α −nδ ≤ C4n+1 n! y− ∞ |yi |

∏ |ym |−1 .

m=1

It can be shown analogously for all n ∈ N, 0 = y ∈ Rd :

n

∂x k(x, y) ≤ Cn+1 n! y−α |yi |−n 4

i

∞

d

∏ |ym |−1 ,

(4.8)

m=1

which completes the proof.

We will need these estimates later to prove exponential convergence of the numerical quadrature rules employed to approximate the discretized generator of the Feller process.

4.2 Sector Condition The sector condition for the symbol of the Feller process will be one of the main ingredients for proving well-posedness of the initial boundary value problems for the PIDEs arising in option pricing problems. We shall require the symbol a(·, ·) to satisfy a sector condition of the following form: ∃ γ > 0 s.t. ∀x, ξ ∈ Rd :

ℜ a(x, ξ ) + 1 ≥ γ ξ m(x) .

(4.9)

Remark 4.3. Note that (4.9) and (3.5) imply the usual sector condition ∃C > 0

s.t. ∀x, ξ ∈ Rd : ℑa(x, ξ ) ≤ Cℜa(x, ξ ),

(4.10)

152


cf. [8]. For constant symbols a(x, ξ ) = ψ (ξ ), arising in Lévy models, the sector condition (4.10) implies that the corresponding bilinear form is a nonsymmetric Dirichlet form, cf. [8, Theorem 3.7] and [51, Chapter 4.7]. Verification of the sector condition is not straightforward for a general Feller process. Here, we give sufficient conditions for the sector condition to hold in terms of appropriate conditions on the marginals of the Feller process and copula function. d

Definition 4.4. The function F : R → R is a homogeneous Lévy copula of order 1. The functions k10 , . . . , kd0 are densities of jump measures of univariate Feller processes of order −1 − m1, . . . , −1 − md , i.e., k0j (x j , ry j ) = r−1−m j (x) k0j (x j , y j ),

∀r > 0 and all x j ∈ R, y j ∈ R\{0}

for any j = 1, . . . , d and F and k0j (x j , y j ), j = 1, . . . , d, satisfy the assumptions of Theorem 4.1. Due to Theorem 4.1 there exists a unique Feller process with corresponding margins. We call such a d-variate Feller process m(x)-stable, for m(x) = (m1 (x1 ), . . . , md (xd )). For the pure jump case we will need the following additional property in order to prove a simple equivalence for the sector condition. We assume that the symmetric part of the jump measure ksym (x, y) = 12 (k(x, y) + k(x, −y)) admits the following estimate: ksym (x, y) k0,sym (x, y),

∀0 < y < 1, ∀x ∈ Rd ,

(4.11)

where k0 is the jump measure of an m(x)-stable Feller process. We will now prove an anisotropic homogeneity property of the Feller density k0 . Theorem 4.5. Let the copula F and the marginal densities be as in Definition 4.4. Then the function k0 given by (4.4) is m-homogeneous in the sense that 1 − m 1(x ) 1+ 1 +···+ m (x − m 1(x ) 0 d d d d ) k (x, y1 , . . . , yn ) 1 1 k x,t y1 , . . . ,t yn = t m1 (x1 ) ∀t > 0. 0

Proof. The proof follows analogously to [38, Theorem 3.2].

Theorem 4.6. Let k0 (x, y1 , . . . , yd ) be as in the previous theorem. Then the Feller 0 symbol (0, 0, k (x, y1 , . .. , yn )) is a real-valued anisotropic homogeneous function of type

1 1 m1 (x1 ) , . . . , md (xd ) 1

and order 1 for all x ∈ Rd , i.e., it satisfies 1

a(x,t m1 (x1 ) ξ1 , . . . ,t md (xd ) ξn ) = ta(x, ξ1 , . . . , ξd ) ∀t > 0, ξ ∈ Rd Proof. The proof follows analogously to [38, Theorem 3.3] using Theorem 4.5. We will need the following Lemma, which is a modification of [30, Lemma 2.2].


153

Lemma 4.7. Let ρ1 (x, y) and ρ 2 (y) ≤ ρ2 (x, y) ≤ ρ 2 (y) be two anisotropic distance functions of order 1 and type m(x) = (m1 (x), . . . , md (x)) for all x ∈ Rd , and let ρ 2 (y), ρ 2 (y) be continuous. Furthermore, let Σ := x∈Rd Σ1 (x), where

Σ1 (x) := {z : ρ1 (x, z) = 1}, is contained in a compact set. Then the following inequalities hold with constants C1 , C2 > 0 independent of x and y: C1 ρ1 (x, y) ≤ ρ2 (x, y) ≤ C2 ρ1 (x, y). Proof. Let y ∈ Rd . We set t(x) =

1 ρ1 (x,y) .

Then

t(x)m1 (x) y1 , . . . ,t(x)md (x) yd ∈ Σ1 (x) holds. As Σ is contained in a compact set and ρ 2 (y), ρ 2 (y) are continuous, we obtain C1 ≤ ρ2 (x, y) ≤ C2

∀x ∈ Rd , ∀y ∈ Σ1 .

Hence, C1 ≤

1 ρ2 (x, y) = t(x)ρ2 (x, y) = ρ2 (x,t(x)m1 (x) y1 , . . . ,t(x)md (x) yd ) ≤ C2 . ρ1 (x, y)

Theorem 4.8. Let X be a Feller process taking values in Rd with characteristic triplet (γ (x), Q(x), k(x, y)dy) with density k(x, y) of the jump-measure constructed parametrically as in Theorem 4.1. Further assume that either Q > 0 holds or that k(x, y) satisfies (4.11) with an m(x)-stable function k0 (x, y). Then, there exists a constant C > 0 such that for all x ∈ Rd and ξ ∞ sufficiently large d

|ℜa(x, ξ )| ≥ C ∑ |ξ |m j (x j ) ,

(4.12)

j=1

where m j (x j ) = 2 in the case Q > 0. Proof. The proof mainly follows the arguments of [93, Proposition 2.4.3]. First consider Q = 0. Due to Theorem 4.6 one obtains that ℜa0 (x, ξ ) is an anisotropic distance function of order m 1(x ) , . . . , m 1(x ) for all x ∈ Rd . Since all anisotropic 1

1

d

d

distance functions of the same order are equivalent there exists some C1 (x) such that d

a0 (x, ξ ) ≥ C1 (x) ∑ |ξi |mi (xi ) , i=1

∀ξ ∈ Rd .

154


Hence, |ℜa(x, ξ )| =

Rn

≥ C2

(1 − cos(ξ · y)) ksym (x, y)dy

B1 (0)

(1 − cos(ξ · y))k0,sym (x, y)dy d

≥ C2C1 (x) ∑ |ξi |mi (xi ) − C3 . i=1

To complete the proof, we must prove the boundedness of C1 (x), i.e., we have to validate the conditions of Lemma 4.7. The compactness of Σ1 follows from the definition of ρ1 (x, y) = ∑di=1 |yi |mi (x) , and the estimates on ρ2 (x, y) follow from the conditions imposed on k0 . Therefore, the sector condition (4.9) follows from (4.11) for a certain set of symbols. The case Q > 0 is trivial.

Assumption (4.11) is implied by the following conditions on the marginal jump measures and the copula function: Assumption 4.9 Let X be a Feller process with characteristic triplet (γ , Q(x), k(x, y)dy) satisfying the conditions of Theorem 4.1. Let the following inequalities hold, with F 0 being a 1-homogeneous Lévy copula as in Assumption 4.4 and ki0 (x, y) being m(x)-stable densities with tail integrals Ui0 (x, y), i = 1, . . . , d: ki (x, y) ki0 (x, y),

∀0 < |y| < 1, ∀x ∈ Rd ,

i = 1, . . . , d ∂1 . . . ∂d F(U(x, y)) ∂1 . . . ∂d F (U 0 (x, y)) ∀0 < |y| < 1. 0

4.3 A Class of Admissible Market Models We now formulate the requirements for market models which will be admissible for our pricing schemes in terms of the marginals and the copula function. These requirements will not only ensure existence and uniqueness of a solution of the corresponding pricing problem, but also ensure that the presented FEM based algorithms are feasible. Definition 4.10. We call a d-dimensional Feller process with characteristic triplet (γ (x), Q(x), N(x, dy)) an admissible market model if it satisfies the following properties. 1. The function x → γ (x) ∈ Rd is smooth and bounded. 2. The function x → Q(x) ∈ Rd×d sym is smooth and bounded and Q(x) is positive d semidefinite for all x ∈ R . 3. The jump measure N(x, dy) is constructed from d independent, univariate Feller-Lévy measures with a 1-homogeneous copula function F that fulfills


155

the following estimate: there is a constant C > 0 such that for all u ∈ (R\{0})d and all n ∈ Nd0 holds d

|∂ n F(u)| ≤ C|n|+1 |n|! min{|u1 | , . . . , |ud |} ∏ |ui |−ni . i=1

4. For the marginal densities Ni (xi , dyi ) = ki (xi , yi )dyi the mapping xi →

B∩R\{0}

(1 ∧ |yi |2 )Ni (xi , dyi ) is smooth for all B ∈ B(R). 5. There exist univariate Lévy kernels ki (y), i = 1, ..., d with semiheavy tails, i.e., which satisfy ki (y) ≤ C

e− β

− |y|

+ e− β y ,

, y < −1 y > 1,

(4.13)

for some constants C > 0, β − > 0 and β + > 1. These Lévy kernels satisfy the following estimates

2

B∩R\{0}

(1 ∧ |yi | )ki (xi , yi )dyi ≤

B∩R\{0}

(1 ∧ |yi |2 )ki (yi )dyi < ∞ ∀xi ∈ R,

for B ∈ B(R), i = 1, . . . , d. 6. Besides, we require the following estimate on the derivatives of ki (x, y)

n

∂x ki (xi , yi ) ≤ Cn+1 n! |yi |−mi (xi )−δ n−1 , i

n

∂y ki (xi , yi ) ≤ Cn+1 n! |yi |−mi (xi )−n−1 , i for any δ ∈ (0, 1), for all 0 = y, x ∈ Rd and m := maxi=1,...,d supxi ∈R mi (xi ) < 2 as well as m := mini=1,...,d infxi ∈R mi (xi ) > 0. 7. Finally we require F 0 to be a 1-homogeneous Lévy copula and ki0 (xi , yi ) to be mi (xi )-stable densities with tail integrals Ui0 (xi , yi ), i = 1, . . . , d: ki (xi , yi ) ki0 (xi , yi ), ∀0 < |yi | < 1, ∀xi ∈ R, i = 1, . . . , d ∂1 . . . ∂n F(U(x, y)) ∂1 . . . ∂n F 0 (U 0 (x, y)) ∀0 < |y| < 1. Remark 4.11. An admissible market model satisfies the requirements of Theorem 4.1 due to conditions (3), (5) and (6). Lemma 4.12. The symbol a(x, ξ ) of an admissible market model X with triplet m(x) (0, Q(x), k(x, y)dy) is contained in the symbol class S1,δ for any δ ∈ (0, 1), where m(x) = 2 if Q(x) ≥ Q0 > 0. Prior to the proof of the preceding lemma we remark that the removal of the drift will be discussed in the next chapter. The proof follows analogously to

156


[73, Proposition 3.5]. Let us illustrate the preceding, abstract developments with an example related to the so-called tempered-stable class of Lévy processes which were advocated in recent years in the context of financial modelling. Example 4.13 (Feller-CGMY). We consider a d-dimensional Feller process with Clayton Lévy copula F(u1 , . . . , ud ) = 22−d

d

∑ |ui|ϑ

− 1

ϑ

ρ 1{u1 ,...,ud ≥0} − (1 − ρ )1{u1,...,ud ≤0} ,

i=1

where ϑ > 0, ρ ∈ [0, 1] together with CGMY-type densities ki (x, y) = C(x)

−

e− β i

(x)|y|

|y|1+mi (x)

+

1{y0} ,

with smooth and bounded functions C(x) > 0, βi− (x) > 0, βi+ (x) > 1, 0 < mi < mi (x) ≤ mi < 2, for i = 1, . . . , d. We assume the Gaussian component Q(x) to be positive semidefinite, smooth and bounded. The drift γ (x) is assumed to be smooth and bounded. It is easy to see that this market model satisfies Properties (1), (2), (4)–(6) of the above definition. (3) and (7) follow analogously to the proof of [93, Proposition 2.3.7].

5 Variational PIDE Formulations A key observation at the heart of differential equation approaches to deterministic computational pricing of derivative contracts in finance is the observation (going back at least to R. Feynman and M. Kac) that conditional expectations over all sample paths of a multivariate diffusion process satisfy deterministic, parabolic partial differential equations (PDEs). The most well known representative of these PDEs in financial modelling is the classical Black-Scholes equation. This Feynman Kac correspondence holds in a much more general context, the deterministic equation being in general nonlinear, and the solution being in general understood as viscosity solution. Here, we will follow on linear differential equations for which the (unique) solutions will be variational solutions of suitable weak formulations of the deterministic evolution equations. As these formulations will form the basis of variational discretizations to be discussed below, we shall present their ingredients (Sobolev and Besov spaces, Dirichlet Forms, Evolution triplets and the abstract theory of parabolic evolution equations) in some detail here. Throughout, our perspective is the pricing of derivative contracts in financial models based on the Feller-Lévy processes introduced above.


157

5.1 European Options We consider a European option with maturity T < ∞ and payoff g(ST ) which is i assumed to be Lipschitz, where Sti = S0i ert+Xt and where X is a semimartingale unless specified otherwise. By the general theory of asset pricing (as, e.g., in [31]), an arbitrage free value V (t, s) of this option is given by V (t, s) = E e−r(T −t) g(ST )|St = s , where the expectation is taken under the measure Q which is equivalent to the real world measure and under which ST is a sigma-martingale, cf. [31]. If X is an admissible market model, we can derive a PDO and PIDE representation and prove well-posedness of the weak formulation of the problem on a bounded domain. If X is a (multidimensional) continuous semimartingale, satisfying certain non-degeneracy conditions, we can use a mimicking result due to [41] and derive a state-space and time inhomogeneous PDE. Existence and uniqueness on bounded and unbounded domains are well-known under certain smoothness and growth conditions on the coefficients. If X is a (multidimensional) discontinuous semimartingale satisfying certain non-degeneracy conditions, we can use a novel mimicking result due to [6] and derive a state-space and time dependent PDO and PIDE. Existence and uniqueness results are not yet available in this case. In the following we will concentrate on the time homogeneous Feller case. Due to no arbitrage considerations we will require the considered processes to be martingales under a pricing measure Q. This requirement can be expressed in terms of the characteristic triplet: Lemma 5.1. Let X be constructed according to Theorem 4.1 with characteristic triplet (γ (x), Q(x), N(x, dy)) and semigroup (Tt )t≥0 . Further let Tt (ex j ) < ∞ hold for t ≥ 0 and the processes Xj with characteristic triplet (0, 0, 1{x j >1} N j (x j , dy j )), j = 1, . . . , d, be processes with bounded variation. Then eX j is a Q-martingale with respect to the canonical filtration of X if and only if Q j j (x)2 + γ j (x) + 2

0 =y∈R

(ey j − 1 − y j )N j (x, dy j ) = 0 ∀x ∈ R.

Proof. This is a direct consequence of [37, Section 3].

(5.1)

Remark 5.2. Note that without assuming finiteness of exponential moments of the processes X j , the processes eX j , j = 1, . . . , d would generally only be a local martingale. For Lévy processes exponential decay of the jump measure implies the existence of exponential moments, cf. [76, Theorem 25.3], this is not obvious for general Feller processes. Recently [59] have proven finiteness of exponential moments for a certain class of Feller processes assuming exponential decay of the density of the jump measure.

158


We are now able to derive a PDO and PIDE for option prices. Let the stochastic process X be an admissible markovian market model and let be g ∈ V := D(AX ), where V = H 1 (Rd ) for diffusion market models V = H m (R) m = [m1 , . . . , md ] ∈ [0, 1]d for general Lévy models and V = H m(x) , with m(x) as in Definition 3.2, for Feller market models considered here. Then we obtain due to semigroup theory for u(t, x) = Tt (g) = E[g(Xt )|X0 = x], where we have set t0 = 0 and r = 0 for notational convenience by differentiation in t:

∂t u − AX u = 0 in (0, T ) × Rd u0 = g in {0} × Rd .

(5.2) (5.3)

Testing with a function v ∈ V , we end up with the following parabolic evolution problem: Find u ∈ L2 ((0, T ); V )∩H 1 ((0, T ); V ∗ ) s.t. for all v ∈ V and a.e. t ∈ [0, T ] holds u0 = g, (5.4) (∂t u, v) − a(u, v) = 0, where the bilinear form a(u, v) = (AX u, v) is the Dirichlet form of the stochastic process X. Although in option pricing, only the homogeneous parabolic problem (5.4) arises, the inhomogeneous equation (5.5) is useful in many applications. We mention only the computation of the time-value of an option, or the computation of quadratic hedging strategies and the corresponding hedging error. Thus, we will in general consider the nonhomogeneuos analogon of the above equation. The general problem reads: Find u ∈ L2 ((0, T ); V ) ∩ H 1 ((0, T ); V ∗ ) s.t. (∂t u, v) − a(u, v) = ( f , v)V ∗ ×V u0 = g

in (0, T ), ∀v ∈ V

(5.5)

for some f ∈ L2 ((0, T ); V ∗ ). Now we consider the localization of the unbounded problem to a bounded domain D. To be able to control the error introduced by localization we need to require the following growth condition on the payoff function: There exists some q ≥ 1 such that q g(s)

d

∑ si + 1

,

∀s ∈ Rd≥0 .

(5.6)

i=1

For any function u with support in a bounded domain D ⊂ Rd we denote by u the zero extension to Rd and define AD (u) = A ( u) with domain VD . The variational formulation of the pricing equation on a bounded domain D ⊂ Rd reads: Find u ∈ L2 ((0, T ); VD ) ∩ H 1 ((0, T ); (VD )∗ ) s.t. for all v ∈ VD and a.e. t ∈ [0, T ] holds: (∂t u, v) − aD (u, v) = ( f , v)VD∗ ×VD u0 = g|D ,

(5.7) (5.8)

u, v) and u, v denote the zero extensions of u and v to Dc = where aD (u, v) := a( d R \D. Note that the spaces VD =: {v ∈ L2 (D) : v ∈ V } consist of functions which


159

vanish in a weak sense on ∂ D. Under condition (5.6) pointwise convergence of the solution of the localized problem to the solution of the original problem can be shown for Lévy processes using [76, Theorem 25.18] and the semiheavy tail property. We refer to [73, Theorem 4.14] for details. A comparable result for general Feller processes does not appear to be available yet. Remark 5.3. This formulation naturally arises for payoffs with finite support such as digital or (double) barrier options. The truncation to a bounded domain can thus be interpreted economically as the approximation of a standard derivative contract by a corresponding barrier option on the same market model. Note also that the variational framework (5.7)–(5.8) naturally allows for more general initial conditions, in particular g ∈ H = L2 (Rd ). Therefore, discontinuous g are admissible in the variational framework (5.7)–(5.8). This is essential for the pricing of exotic contracts such as digital options, for example. Existence and uniqueness of weak solutions of (5.7)–(5.8) follows analogously to [84, Section 6.2], so we obtain the following theorem: 2m(x)

with m(x) = (m1 (x1 ), . . . , Theorem 5.4. Let the generator A (x, D) ∈ Ψρ ,δ md (xd )) be a pseudo-differential operator of variable order 2m(x), 0 < mi (xi ) < 1, 2m(x) i = 1, . . . , d with symbol a(x, ξ ) ∈ Sρ ,δ for some 0 < δ < ρ ≤ 1 for which there exists γ > 0 with ∀x, ξ ∈ Rn :

ℜ a(x, ξ ) + 1 ≥ γ ξ 2m(x) .

(5.9)

2m(x)

Then A (x, D) ∈ Ψρ ,δ satisfies a G˚arding inequality in the variable order space m(x) (D): There are constants γ > 0 and C ≥ 0 such that H m(x) (D) : ∀u ∈ H

ℜa(u, u) ≥ γ u2H m(x) (D) − Cu2L2(D) ,

(5.10)

and ∃λ > 0

m(x) (D) → H −m(x) (D) such that A (x, D) + λ I : H

(5.11)

is boundedly invertible. As noted above, we obtain H˜ 1 (D) = H01 (D) as the domain of the operator if Q(x) ≥ Q0 > 0, i.e. if the diffusion matrix Q(x) is uniformly positive definite. Theorem 5.5. The problem (5.7)–(5.8) for an admissible market model X with initial condition gD ∈ H has a unique solution. Proof. Using the G˚arding inequality (5.10) and the continuity of the operator the result follows from standard theory of parabolic evolution equations.

Remark 5.6. We obtain for finite variation pure jump models, i.e., m(x) < 0.5, for all x ∈ D, an advection dominated equation. Therefore we have to remove the drift

160


for standard algorithms to be feasible. This is easy in the Lévy case as the drift coefficients in the equation are constant, cf. [73, Corollary 4.3], but more involved in the Feller case, cf. [43]. Theorem 5.7. For f ∈ C1,2 (J × R), with ∂x f (t, x) = 0, consider the change of variable v(t, x) := u(t, f (t, x)), where u(t, x) is the solution of the following PDE

∂t u − Σ (x)∂xx u + b(x)∂x u + c(x)u = 0 in J × R. Let f solve the (nonlinear) PDE

∂t f − Σ ( f (t, x))

∂xx f − b( f (t, x)) = 0. ∂x2 f

(5.12)

Then v satisfies the PDE

∂t v −

Σ ( f (t, x)) ∂xx v + c( f (t, x))v = 0 in J × R. ∂x2 f

Solving the PDE (5.12) is non trivial in general. Remark 5.8. We come back to the CEV model introduced in Remark 2.10. The generator A CEV is given as 1 A CEV u(x) = σ 2 x2ρ ∂xx u(x) + rx∂x u(x) 2 and the corresponding bilinear form aCEV on a bounded domain G reads: 1 aCEV (u, v) = σ 2 2

G

x2ρ ∂x u∂x v dx +

G

(ρσ 2 − rx)∂x uv dx +

uv dx. G

The domain of the generator is the weighted Sobolev space Hρ , defined as the Hρ := C0∞ (G)·ρ , where u2ρ

=

G

x2ρ |∂x u|2 + |u|2 dx.

A G˚arding inequality and continuity of the bilinear form aCEV can be shown on Hρ .

5.2 American Options For the study of optimal stopping problems which arise e.g. from American contracts we require variational formulations of parabolic variational inequalities.


161

To this end, let 0/ = K ⊂ V be a closed, non-empty and convex subset of V with indicator function

φ (v) := IK (v) =

0,

if v ∈ K ,

+ ∞,

else.

(5.13)

This is a proper, convex lower semicontinuous (l.s.c.) function φ : V → R with ◦ domain D(φ ) = {v ∈ V : φ (v) < ∞}. We denote by K H the closure of D(φ ) in H and consider the following variational problem: Given f ∈ L2 (0, T ; V ∗ ), u0 ∈ K

◦H

⊂H ,

find u ∈ L2 (0, T ; V ) ∩ H 1 (0, T ; V ∗ ) such that u ∈ D(φ ) a.e. in (0, T ) and ∂t u + A u − f , u − vV ∗ ×V + φ (u) − φ (v) ≥ 0 ∀v ∈ D(φ ), a.e. in (0, T ), (5.14) u(0) = u0

in

H.

(5.15)

Existence and uniqueness results for solutions u ∈ L2 (0, T ; V ) of (5.14)–(5.15) can be obtained from e.g. [40, Theorem 6.2.1] under rather strict conditions on the data f . To derive the well-posedness of (5.14)–(5.15) under minimal regularity conditions on f , u0 and φ , the problem needs to be replaced by a weak variational formulation. To state it, we introduce the integral functional Φ on L2 (0, T ; V )

Φ (v) =

⎧ ⎨ ⎩

0

T

φ (v(t))e−2λ t dt,

+ ∞,

if φ (v) ∈ L1 (0, T ),

(5.16)

else,

with λ ≥ 0 as in (5.10). Note that Φ (·) is proper convex and l.s.c. with domain D(Φ ) = {v ∈ L2 (0, T ; V ) : φ (v) ∈ L1 (0, T )}.

(5.17)

Herewith, the weak variational formulation of (5.14)–(5.15) reads (cf. [3,77]): Given ◦ u0 ∈ K H ⊂ H and f ∈ L2 (0, T ; V ∗ ), find u ∈ L∞ (0, T ; H ) ∩ D(Φ ) such that u(0) = u0 in H and T 0

∂t v(t) + (A + λ )u(t) − ( f + λ v), u(t) − v(t) · e−2λ t dt + Φ (u) − Φ (v) (5.18)

1 ≤ u0 − v(0)2H , 2 for all v ∈ D(Φ ) with ∂t v ∈ L2 (0, T ; V ∗ ). The well-posedness of (5.18) is ensured by [77, Theorem 3]:

162


Theorem 5.9. Assume that the infitesimal generator AX is coercive and continuous. Then problem (5.18) admits a unique solution u ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H )

such that t → φ (u(t, ·)) ∈ L1 (0, T ).

Remark 5.10. As for the parabolic equality problem, also for (5.18) the initial condition is only required to hold in H . In addition, however, in (5.18) the data u0 must ◦ belong to the closure K H of K in H . Remark 5.11. Convergence rates of backward Euler time discretizations of (5.18) for American style contracts under minimal regularity are given in [3, 65, 77]. Using the notation of the previous sections, we now consider an American option with maturity T < ∞ and Lipschitz continuous payoff g(S). Its price VA (t, S) is given by the optimal stopping problem VA (t, S) = sup E e−r(T −τ ) g(Sτ )|St = S , (5.19) τ ∈Tt,T

where Tt,T denotes the set of all stopping times between t and T . i In [66, 67] it is shown how the price VA (t, S) for Sti = S0i ert+Xt , X being a Lévy process, can be characterized as the viscosity solution of a corresponding Bellman equation (for details on viscosity solutions we refer to e.g. [27] and the original sources [28, 78, 89]): Theorem 5.12. The price VA (t, S) of an American option defined in (5.19) is a viscosity solution of min −∂t VA (t, S) − rVA (t, S) −

d 1 d Si S j Qi j ∂S2i S j VA − r ∑ Si ∂Si VA (t, S) ∑ 2 i, j=1 i=1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ d z zi − VA (t, Se ) −VA (t, S) − ∑ Si (e − 1) ∂Si VA (t, S) N( dz),VA (t, S) − g(S) ⎪ Rd ⎪ i=1 ⎪ ⎪ ⎪ ⎭ AJVA

= 0.

(5.20)

If VA (t, S) is uniformly continuous and there holds sup [0,T ]×Rd>0

VA (t, S) < ∞, 1+S

(5.21)

this solution is unique. Proof. Existence of the viscosity solution follows from [67, Theorems 3.1] and its uniqueness is ensured by [67, Theorems 4.1] and [78].


163

Remark 5.13. Note that Theorem 5.12 holds only in the Lévy case. The solvability of the Bellman equation for more general jump measures is investigated in [1, 18]. The Bellman equation (5.20) can equivalently be restated as the following linear complementarity problem:

∂τ uA (τ , x) + ABS [uA ](τ , x) + AJ [uA ](τ , x) = l(τ , x) ≤ 0, uA (τ , x) − erτ gτ (x) ≥ 0, l(τ , x) uA (τ , x) − erτ gτ (τ , x) = 0,

(5.22)

on [0, T ] × Rd with ABS and AJ as above. The initial condition is given by uA,0 = g(ex ), i.e. uA,0 = u0 . The function gτ is the transformed payoff function, where we applied a transformation as in Theorem 5.7. The system (5.22) can also be considered for Feller generators and on bounded domains. An analogous localization argument to the European case can be used for Lévy market models. Furthermore, if the solution uA of (5.22) satisfies uA ∈ L2 ((0, T ); VD ) ∩ H 1 ((0, T ); D(AD )∗ ) it can be identified with the solution of the following realization of the abstract variational inequality (5.14)–(5.15): Find uA ∈ L2 ((0, T ); VD ) ∩ H 1 ((0, T ); VD∗ ) such that uA ∈ D(φτ ) a.e. in (0, T ) and ∂τ uA , v − uAVD∗ ,VD + (AD uA , v − uA) − φτ (u) + φτ (v) ≥ 0 , (5.23) for all v ∈ D(φτ ), a.e. in (0, T ), and uA (0) = u0 ,

with φτ := IKτ as in (5.13) and convex sets Kτ := {v ∈ VD : v ≥ erτ gτ } ⊂ VD ,

τ ∈ (0, T ),

where gτ : Rd → R as above. In weak form the variational problem (5.23) reads: Find uA ∈ L∞ ((0, T ); VD ) ∩ H 1 ((0, T ); VD∗ ) such that uA ∈ D(Φ )a.e. in (0, T ) and T 0

∂τ v(τ ) + (AD + λ )uA (τ ) − λ v(τ ), uA (τ )

−v(τ ) · e−2λ τ d τ + Φ (uA ) − Φ (v) 1 ≤ u0 − v(0)2H , 2 for all v ∈ D(Φ ) with ∂τ v ∈ L2 (0, T ; V ∗ ).

(5.24)

164


Here Φ and D(Φ ) are depending on φτ as defined in the last section. The wellposedness of (5.24) is ensured by Theorem 5.14. Let X be a Feller process which is an admissible market model with state space Rd , characteristic triplett (γ , Q, N) and infinitesimal generator A. Then the weak variational inequality (5.24) with u0 ∈ L2 (D) admits a unique solution in VD . Proof. The proof follows from Theorem 5.9 using the G˚arding inequality (5.10) and continuity of the corresponding Dirichlet bilinear form on VD in conjunction with, e.g., [17, Remarque 3] (to account for the smooth time dependence of the convex set Kτ ).

Remark 5.15. For Lévy processes with a nondegenerate diffusion component, i.e., Q(x) ≥ Q0 > 0 in D we obtain H01 (D) as the domain of the generator. For Lévy processes an analogous result holds, cf. [73, Theorem 4.8]. A different approach to prove existence and uniqueness for the pricing problem is by discretization in time, i.e. to approximate the parabolic variational inequalities by a sequence of nonlinear elliptic equations and prove well-posedness for the solution of each equation in the sequence and apriori estimates for all elements of the sequence. Convergence of the sequence of solutions in an appropriate sense can be proven. This procedure also gives a feasible numerical scheme for the approximation of the pricing problem, for more details we refer to [48, 49] and references therein. To approximate the solution of (5.22) and (5.23) we consider the following oneparameter family of regularized problems: for a regularization parameter c > 0, consider Find ucA ∈ L2 ((0, T ); VD ) ∩ H 1 ((0, T ); VD∗ ) such that ∂τ ucA , v − ucAVD∗ ,VD + (AD ucA , v − ucA) + min 0, l(τ , x) + c(ucA − erτ gτ ) , v − ucA ≥ 0 ,

(5.25)

for all v ∈ V , c > 0, a.e. in (0, T ), and uA (0) = u0 . c

Different choices for the function l are possible, if l ∈ L2 ((0, T ); L2 (D)) we obtain existence of a unique solution of (5.25) for Feller models, cf. [49, Theorem 1]. Under additional assumptions on the bilinear form and the function l(τ , x) convergence of the sequence of solutions (ucA )c>0 to uA with order O(1/c) in the L∞ ((0, T ) × D) norm can be established, cf. [49, Theorem 4.2]. After discretization in time a semi-smooth Newton method can be employed to solve the arising nonlinear systems in each time step iteratively, cf. [48].

5.3 Greeks A key task in financial engineering is the fast and accurate calculation of sensitivities of market models with respect to model parameters. This becomes necessary for


165

example in model calibration, but also in quantification of model uncertainty for risk analysis and in the pricing and hedging of certain derivative contracts. Classical examples are variations of option prices with respect to the spot price or with respect to time-to-maturity, the so-called “Greeks” of the model. For classical, diffusion type models and plain vanilla type contracts, the Greeks can be obtained analytically. With the trends to more general market models of jump-diffusion type and to more complicated contracts, closed form solutions are generally not available for pricing and calibration. Thus, prices and model sensitivities have to be approximated numerically. We will consider the sensitivity of the solution u to variation of a model parameter, like the Greek Vega (∂σ u) and the sensitivity of the solution u to a variation of state spaces such as the Greek Delta (∂x u). Definition 5.16. We call a process X a parametric Feller market model with admissible parameter set Sη , if the mapping Sη η → {γ , Q, N} is infinitely differentiable. For a parametric Markovian market model X in the sense of Definition 5.16 we distinguish two classes of sensitivities. In the following we assume that X(η0 ) is an admissible market model i.e. AD (η0 ) is continuous and satisfies the G˚arding inequality for all η0 ∈ Sη . Note that the domain VD of the operator AD might depend on η0 . For the numerical computation of sensitivities as well as for quadratic hedging it will be crucial to admit a non-trivial right hand side. Accordingly, we consider the parabolic problem

∂t u − AD(η0 )u = f

in J × D,

u(0, x) = u0 in D,

(5.26)

with u0 = g. For f ∈ L2 (J; VD∗ ) and u0 ∈ H the weak formulation of the problem (5.26) is given by: Find u ∈ L2 (J; VD ) ∩ H 1 (J; H ) such that (∂t u, v) − a η0 ; u, v = f , vVD∗ ,VD , u(0, ·) = u0 .

∀v ∈ VD ,

(5.27)

Under the assumption that continuity and the G˚arding inequality hold for every model parameter η0 ∈ Sη , the problem (5.27) admits a unique solution. We will distinguish two classes of sensitivities: 1. The sensitivity of the solution u to a variation Sη ηs := η0 + sδ η , s > 0, of a model parameter η0 ∈ Sη . Typical examples are the Greeks Vega (∂σ u), Rho (∂r u) and Vomma (∂σ σ u). Other sensitivities which are not so commonly used in the financial community are the sensitivity of the price with respect to the jump intensity or the order of the process that models the underlying. 2. The sensitivity of the solution u to a variation of arguments t, x. Typical examples are the Greeks Theta (∂t u), Delta (∂x u) and Gamma (∂xx u).

166


We note that Gamma ∂xx u and Vomma ∂σ σ u are second derivatives of u. The most straightforward approach to their numerical computation is to first obtain a numerical approximation u˜ of u and then to differentiate u˜ with respect to the respective parameters. In variational discretizations such as the ones we will introduce below, u˜ will be a continuous, piecewise linear function of x. Therefore, direct computation of the Gamma ∂xx u by differentiation of u˜ is meaningless (it will yield a finite combination of Dirac distributions) and a more sophisticated approach, based on postprocessing the approximate variational solution u˜ by averaging is required. Likewise, the stable numerical computation of sensitivities with respect to a model parameter is based on the observation that the sensitivity of interest satisfies (5.5) with a suitable f .

5.3.1 Sensitivity with Respect to the Model Parameter Let C be a Banach space over a domain D ⊂ Rd . C is the space of parameters or coefficients in the operator A and Sη ⊆ C is the set of admissible coefficients. We denote by u(η0 ) the unique solution to (5.27) and introduce the derivative of u(η0 ) with respect to η0 ∈ Sη as the mapping Dη0 u(η0 ) : C → VD 1 u(η0 + sδ η ) − u(η0) , s→0+ s

u(δ η ) := Dη0 u(η0 )(δ η ) := lim

δη ∈ C .

We also introduce the derivative of AD (η0 ) with respect to η0 ∈ Sη D (δ η )ϕ := Dη A (η0 )(δ η )ϕ := lim 1 AD (η0 + sδ η )ϕ − AD (η0 )ϕ , A 0 s→0+ s D (δ η ) ∈ L (V D , V D ∗ ) with V D a real and where ϕ ∈ VD , δ η ∈ C . We assume that A separable Hilbert space satisfying VD ⊆ VD ⊂ H ∼ = H ∗ ⊂ VD∗ ⊆ VD∗ . D such We further assume that there exists a real and separable Hilbert space VD ⊆ V ∗ that AD v ∈ VD , ∀v ∈ VD . We have the following relation between Dη0 u(η0 )(δ η ) and u. D (δ η ) ∈ L (V D , V D ∗ ), ∀δ η ∈ C and u(η0 ) : J → VD , η0 ∈ Sη Lemma 5.17. Let A be the unique solution to

∂t u(η0 ) − AD (η0 )u(η0 ) = 0 in J × D,

u(η0 )(0, ·) = g(x) in D.

(5.28)

Then, u(δ η ) solves D (δ η )u(η0 ) in J × D, ∂t u(δ η ) − AD (η0 ) u(δ η ) = A

u(δ η )(0, ·) = 0 in D. (5.29)


167

Proof. Since u(η0 )(0) = g does not depend on η0 its derivative with respect to η is 0. Now let ηs := η0 +sδ η , s > 0, δ η ∈ C . Subtract from the equation ∂t u(ηs )(t)− AD (ηs )u(ηs )(t) = 0 equation (5.28) and divide by s to obtain

∂t

1 1 u(ηs )(t) − u(η0 )(t) − AD (ηs ) − AD (η0 ) u(ηs )(t) s s 1 − AD (η0 ) u(ηs )(t) − u(η0 )(t) = 0. s

Taking lims→0+ gives equation (5.29).

D × V D → R D (δ η ) the bilinear form a(δ η ; ·, ·) : V We associate to the operator A which is given by D (δ η )u, v ∗ . a(δ η ; u, v) = A V ,V D

D

The variational formulation to (5.29) reads: Find u(δ η ) ∈ L2 (J; VD ) ∩ H 1 (J; H ) such that (∂t u(δ η ), v)H − a η0 ; u(δ η ), v = + a δ η ; u(η0 ), v , ∀v ∈ VD , (5.30) u(δ η )(0) = 0 . Note that (5.30) has a unique solution u(δ η ) ∈ VD due to the assumptions on D and u(η0 ) ∈ VD . The numerical solution of (5.30) will be discussed a(η0 ; ·, ·), A in the next chapters.

5.3.2 Sensitivity with Respect to Solution Arguments We discuss the computation of D n u = ∂xn11 · · · ∂xdd u for arbitrary multi-index n ∈ Nd0 , where n = (n1 , . . . , nd ). For μ ∈ Zd and h > 0 we define the translation operator e μ Th ϕ (x) = ϕ (x + μ h) and the forward difference quotient ∂h, j ϕ (x) = h−1 (Th j ϕ (x)− d ϕ (x)), where e j , j = 1, . . . , d, denotes the j-th standard basis vector in R . For n ∈ n n1 · · · ∂h,dd ϕ and by Dhn the difference operator of order Nd0 we denote by ∂hn ϕ = ∂h,1 n≥0 n

Dhn ϕ :=

∑

γ ,|n|=n

γ

Cγ ,n Th ∂hn ϕ .

Definition 5.18. The difference operator Dhn of order |n| = n and mesh width h is called an approximation to the derivative D n of order s ∈ N0 if for any G0 ⊂ G there holds s+r+n (G). D n ϕ − Dhn ϕ Hr (G ) ≤ Chs ϕ Hs+r+n (G) , ∀ϕ ∈ H 0

(5.31)

168


Using finite elements for the discretization with basis b1 , . . . , bN of VN , the action of Dhn to vN ∈ VN can be realized as matrix-vector multiplication vN → Dnh vN , where Dnh = Dhn b1 , · · · , Dhn bN ∈ RN×N , and vN is the coefficient vector of vN with respect to the basis of VN . Example 5.19. Let VN be, the space of piecewise linear continuous functions on [0, 1] vanishing at the end points 0, 1. For α , β , γ ∈ R and μ ∈ N0 we denote by diagμ (α , β , γ ) the matrices ⎛

··· 0 α β γ ⎜ ··· 0 α β diagμ (α , β , γ ) = ⎝ .. .. . .

0 γ .. .

⎞ ··· 0 ··· ⎟ ⎠ .. .. . .

where the entries β are on the μ -th lower diagonal. Then, the matrices Qh of the μ forward difference quotient ∂h and Tμ of the translation operator Th respectively are given by Qh = h−1 diag0 (0, −1, 1),

Tμ = diagμ (0, 1, 0).

Hence, for example, we have for the centered finite difference quotient Dh2 ϕ (x) = h−2 (ϕ (x + h) − 2ϕ (x) + ϕ (x − h)), of order 2 in one dimension D2h = T−1 Q2h = h−2 diag0 (1, −2, 1). In the multidimensional case the matrix Dnh is given by Dnh =

∑

γ ,|n|=n

n

Cγ ,n Tγ1 ⊗ · · · ⊗ Tγd Qnh1 ⊗ · · · ⊗ Qhd .

6 Wavelets For the numerical solution, we discretize the parabolic equation (5.7)–(5.8) in m(x) (D) and (0, T ) × D in the spatial variable with spline wavelet bases for V = H in the time parameter by the θ -scheme or the more sophisticated discontinuous Galerkin timestepping which allows to expoit the time-analycity of the processes’ semigroups. To present the spatial discretizations, we briefly recapitulate basic definitions and results on wavelets from, e.g., [23] and the references there. For specific spline wavelet constructions on a bounded interval I, we refer to e.g. [33, 69] and [91]. Since for all infinitesimal generators arising in connection with Markov processes the Sobolev order 2m(x) of the generator satisfies 0 ≤ m(x) ≤ 1, the full machinery of multiresolution analysis in Sobolev spaces of arbitrary order is not required; we confine ourselves therefore to continuous, piecewise polynomial


169

multiresolution systems in R1 . For wavelet discretizations of Kolmogorov equations for multivariate models, we shall employ tensor products of these univariate, piecewise polynomials multiresolution systems. Our use of compactly supported, piecewise polynomial multiresolution systems (rather than the more commonly employed B-spline Finite Element spaces) for the Galerkin discretization of Kolmogorov equations is motivated by the following key properties of these spline wavelet systems: a) the approximation properties of the multiresolution sytems equal those of the B-spline systems, b) the spline wavelet systems form Riesz bases of the domains of the infinitesimal generators of the Markov processes, thereby allowing for simple and efficient preconditioning of the matrices arising in wavelet representations of the processes’ Dirichlet forms, c) the spline wavelet systems can be designed to have a large number of vanishing moments, thereby allowing for a compression of the wavelet matrix for the jump measure.

6.1 Spline Wavelets on an Interval Our Galerkin discretizations of Kolmogorov equations for Feller processes are based on biorthogonal wavelet bases on a bounded interval I ⊂ R. We recapitulate the basic definitions from, e.g., [23, 91] to which we also refer for further references and additional details, such as the construction of higher order wavelets. Our wavelet systems are two-parameter systems {ψl,k }l=−1,...,∞,k∈∇l of compactly supported functions ψl,k . Here the first index, l, denotes “level” of refinement resp. resolution: wavelet functions ψl,k with large values of the level index are welllocalized in the sense that diam(suppψl,k ) = O(2−l ). The second index, k ∈ ∇l , measures the localization of wavelet ψl,k within the interval I at scale l and ranges in the index set ∇l . In order to achieve maximal flexibility in the construction of wavelet systems (which can be used to satisfy other requirements, such as minimizing their support size or to minimize the size of constants in norm equivalences), we will consider wavelet systems which are biorthogonal in L2 (I), consisting of a primal wavelet system {ψl,k }l=−1,...,∞,k∈∇l which is a Riesz basis of L2 (I) (and which will enter explicitly in the Galerkin discretizations of the Markov processes) l,k }l=−1,...,∞,k∈∇l (which will never be and a corresponding dual wavelet system {ψ used explicitly in our algorithms). Notice that construction of fully L2 (I) orthonormal wavelet systems is feasible, but results in function systems which are either nonpolynomial or have larger supports or fewer vanishing moments. The primal wavelet bases ψl,k span finite dimensional spaces W l := span {ψl,k : k ∈ ∇l } ,

V L :=

L−1 & l=−1

Wl

l = −1, 0, 1 . . .,

170


l,k by and the dual spaces are defined analogously in terms of the dual wavelets ψ l := span {ψ l,k : k ∈ ∇l } , W

VL :=

L−1 &

l W

l = −1, 0, 1 . . .,

l=−1

In the sequel we require the following properties of the wavelet functions to be used on our Galerkin discretization schemes, we assume wlog I = (0, 1). l,k satisfy 1. Biorthogonality: the basis functions ψl,k , ψ l ,k = δl,l δk,k . ψl,k , ψ

(6.1)

2. Local support: the diameter of the support is proportional to the meshsize 2−l , l,k ∼ 2−l . diam supp ψl,k ∼ 2−l , diam supp ψ

(6.2)

3. Conformity: the basis functions should be sufficiently regular, i.e. l ⊂ H δ (I) for some δ > 0 , l ≥ −1. 1 (I) , W Wl ⊂H

(6.3)

' l '∞ l Furthermore ∞ l=−1 W , l=−1 W are supposed to be dense in L2 (I). 4. Vanishing moments: The primal basis functions ψl,k are assumed to satisfy vanishing moment conditions up to order p∗ + 1 ≥ p

ψl,k , xα = 0 , α = 0, . . . , d = p∗ + 1, l ≥ 0,

(6.4)

and for all dual wavelets, except the ones at each end point, one has l,k , xα = 0 , α = 0, . . . , d = p + 1, l ≥ 0. ψ

(6.5)

At the end points the dual wavelets satisfy only l,k , xα = 0 , α = 1, . . . , d = p + 1, l ≥ 0. ψ

(6.6)

We remark that the third condition implies that the wavelets satisfy the zero Dirichlet condition, namely ψl,k (0) = ψl,k (1) = 0; the representation of this boundary condition by the subspace is important in the pricing of barrier contracts. To satisfy the homogeneous Dirichlet condition by the wavelet basis, we sacrifice the vanishing moment property of those wavelets whose supports include the endpoints of I, i.e. x = 0 or x = 1. For example, ψl,0 , l = 0, . . ., at the end point x = 0 (assuming that the localization index k ∈ ∇l enumerates the wavelets in the direction of increasing values of x). A systematic and general construction for arbitrary order biorthogonal spline wavelets is presented in [29]. Sufficiently far apart from the end points of (0, 1), biorthogonal wavelet (e.g. [23] and the references there) bases are used in this approach. In the recent paper [42] a wavelet bases was constructed with slightly smaller support at the end points. Using biorthogonal wavelets in the case p = 1,


171

piecewise linear spline wavelets vanishing outside I = (0, 1) are obtained by simple scaling. The interior wavelets have two vanishing moments and are obtained from the mother wavelet ψ (x) which takes the values (0, − 16 , − 13 , 23 , − 13 , − 16 , 0, 0, 0) at the points (0, 18 , 14 , 38 , 12 , 58 , 34 , 78 , 1) by scaling and translations: ψl,k (x) := 2l/2 ψ (2l−3 x − k + 2) for 2 ≤ k ≤ 2l − 3 and l ≥ 3. At the left boundary k = 1, we use the piece−1 1 1 wise linear function ψleft defined by the nodal values (0, 58 , −3 4 , 4 , 4 , 8 , 0, 0, 0) and ψright (x) = ψleft (1 − x). For additional details we refer to [42]. The following particular system of biorthogonal spline wavelet basis functions are Riesz bases for all constant or variable order Sobolev spaces of order s ∈ [0, 1] (and only these spaces arise as domains of the infinitesimal generators of FellerLévy processes) and have proved efficient for our present applications [63]. They are a biorthogonal system of piecewise linear, continuos polynomial spline wavelets which were optimized for having small support. Their dual wavelets do not permit compact support, but they are nevertheless exponentially decaying, i.e.,

Ψe (x) ≤ C exp(−κ |x|),

κ > 0, x ∈ R.

(6.7)

Note that the dual wavelets never enter the Galerkin discretization schemes explicitly. The biorthogonal wavelets in the case p = 1 are continuous, piecewise linear spline wavelets vanishing outside I = (0, 1) (for general intervals I = (a, b), they are obtained by simple scalings). The interior wavelets have two vanishing moments and are obtained from one piecewise linear, continuous mother wavelet function ψ (x) taking values (0, − 12 , 1, − 12 , 0) at (0, 14 , 12 , 34 , 1) by scaling and translation: ψl,k (x) := 2l/2 ψ (2l−1 x − (2k − 1)2−2) for 1 ≤ k ≤ 2l − 2 and l ≥ 2. The boundary wavelets are likewise constructed from the continuous, piecewise linear functions ψ∗ , with values (0, 1, − 12 , 0) at (0, 14 , 12 , 34 ), and ψ ∗ , taking values (0, − 12 , 1, 0) at ( 14 , 12 , 34 , 1): ψ0l = ψ∗ (2l−1 x) and ψl,2l −1 = 2l/2 ψ ∗ (2l−1 x − 2l−1 + 1) (Figure 1). The following results are known for wavelets satisfying the above requirements (e.g., [23]).

V3 (Nodal basis) 0

ψ1

W0 1

2

ψ

3 1

ψ

ψ2

2

ψ1

W2

W3

1

ψ1

W1

2

ψ2

3 2

ψ

3 3

ψ

2

ψ3

3 4

ψ

3 5

ψ

ψ4

3 6

ψ

3 7

ψ

3 8

Fig. 1 Single-scale space VL and its decomposition into multiscale wavelet spaces W for L = 3 and p = 1

172


s (I), 0 ≤ s ≤ p + 1, and, due to the embeddings H m (I) ⊂ Any function v ∈ H m(x) m m(x) H (I) ⊂ H (I), in particular any function v ∈ H (I) can be represented in the wavelet series ∞ Ml

v=∑

∑ vl,k ψl,k =

l=0 k=1

∑

λ ∈I

v λ ψλ ,

vλ =

I

λ dx. vψ

(6.8)

Here, we used the symbol λ = (l, k) to denote a generic index in the index set I := {λ = (l, k) : l = 0, 1, 2 . . . , k = 1, . . . Ml }. m(x) (I) can be obtained by truncating the Approximations vh of functions v ∈ H wavelet expansion (6.8). In this way, a “quasi-interpolating” approximation operator m(x) (I) → Vh , can be defined by truncating the wavelet expansion, i.e. by Qh : H Qh v =

L−1 Ml

∑ ∑ vl,k ψl,k .

(6.9)

l=0 k=1 l

M L −L For all vh = ∑L−1 l=0 ∑k=1 vl,k ψl,k ∈ Vh = V , h ∼ 2 , there holds the norm equivalence

vh 2H s (I) ∼

L−1 Ml

∑ ∑ |vl,k |2 22ls,

(6.10)

l=0 k=1

for all 0 ≤ s < 32 . This result is sharp in the sense that the norm equivalence fails in the upper limit s = 3/2; spline-wavelet systems consisting of higher order, piecewise polynomials with higher regularity across interval boundaries are known, but are not required in the present context, as the arguments in Dirichlet forms of Feller processes must belong locally to H 1 (Rd ), at best. m(x) (I) was shown in [84, Validity of (6.10) in the variable order spaces H m(x) it holds Theorem 3]. There, it was in particular shown that for u ∈ H u2Hm(x) (I) ∼

∞ Ml

2

∑ ∑ ul,k

22mλ l ,

(6.11)

l=0 k=1

where we recall the notation λ = (l, k) ∈ I and mλ which is defined as mλ := inf{m(x) : x ∈ Ωλ } and mλ := sup{m(x) : x ∈ Ωλ }

(6.12)

for the extended support Ωλ of a wavelet basis function ψλ defined by

Ωλ := Ωl,k = For 0 ≤ s
1, we define the subspace VL of H ) product of d univariate approximation spaces, i.e. VL := 1≤i≤d V li , which can be written as * + VL = ψl,k : 0 ≤ li ≤ L − 1, ki ∈ ∇li , i = 1, . . . , d , with basis functions ψl,k = ψl1 ,k1 · · · ψld ,kd , 0 ≤ li ≤ L − 1, ki ∈ ∇li , i = 1, . . . , d. We can write VL in terms of increment spaces VL =

&

W l1 ⊗ . . . ⊗ W ld .

0≤li ≤L−1

Therefore, we have for any function u ∈ L2 (D) the series representation u=

∞

∑ ∑

li =0 ki ∈∇li

ul,k ψl,k .

Using the one dimensional norm equivalences and the intersection structure we obtain 2mdλ ld 2m1λ l1 2 uH m(x) ∑ 2 1 + . . . + 2 d (6.16) |uλ |2 . λ

174


1 (D) for some 1 ≤ s ≤ p + 1. Then for the quasiCorollary 6.2. Let u ∈ H s (D) ∩ H M li interpolant uh = Qh u = ∑L−1 li =0 ∑k=1 ul,k ψl,k there holds for 0 < m < 1 ≤ s ≤ p + 1 the Jackson estimate 22L(m1 (x1 )−s) + . . . + 22L(md (xd )−s) (|Ds u(x)|2 + |u(x)|2 )dx u − uh2Hm(x) (D) 2

I 2L(m−s)

u2H s (D) ,

where m = maxi=1,...,d mi . Proof. For multi-indices λ = (l, k), μ = (L, k ) ∈ I , we introduce the notation λ μ if li ≥ Li and supp ψλi ∩ supp ψμi = 0/ for all i = 1, . . . , d. For s ≥ 32 we choose s < s with 1 ≤ s < 32 , otherwise we set s = s. We observe that mλi −s ≤ mμi −s < 0 holds for all λi μi . Therefore we conclude from the norm equivalence (6.16) u − uh2Hm(x) (D) ∼ =

Mli

∑∑

li ≥L ki =1 Mli

∑∑

li ≥L ki =1

∑

μ ∈∇L

×

2l md 2l m1 2 1 λ1 + . . . + 2 d λd |uλ |2 2l (md −s ) 2l s 2l (m1 −s ) 2l1 s 2 + . . . + 2 d λd 2 d |uλ |2 2 1 λ1

1 2L(mdμ −s ) d 22L(mμ1 −s ) + . . . + 2

∑

λ μ

22s l1 + . . . + 22s ld |uλ |2 ,

where ∇L = { μ = (L, k ) : ki = 1, . . . , ML , i = 1, . . . , d}. d [2−L k , 2−L (k + 1)]. Then, by the norm Let μ = (L, k ), L = |μ | and 2μ := Πi=1 i i equivalence (6.16) and the approximation property (6.14), we have

∑ ∑

μ ∈∇L λ μ

22s l1 + . . . + 22s ld |uλ |2

i

∑

μ ∈∇L

22L(s −s) u2H s (2μ ) .

i

Recalling that 2Lmμ ∼ 2Lm (xi ) ∼ 2Lmμ holds for all x ∈ 2μ , we obtain the final result u − uh2Hm(x) (D)

i

22L(m1 (x1 )−s) + . . . + 22L(md (xd )−s) (|Ds u(x)|2 + |u(x)|2 )dx

I 2L(m−s)

2

u2H s (D) .

6.3 Space Discretization For computational reasons it is convenient to consider the PIDE formulation of the Feller generator AX with symbol a(x, ξ ). For u ∈ S(Rd ) we can write as in (2.7):


AX u(x) =

1 (2π )d

Rd

175

a(x, ξ )u( ˆ ξ ) dξ d

= −γ (x) · ∇u(x) + ∑ Qkl (x)∂kl u k,l

1 + (2π )d

Rd

e

iξ ·x

1−e

Rd

iy·ξ

+

iy · ξ

N(x, dy)u( ˆ ξ ) dξ

1 + |y|2

d

= −γ (x) · ∇u(x) + ∑ Qkl ∂kl u k,l

1 + (2π )d

Rd

Rd

e

iξ ·x

1−e

iy·ξ

+

iy · ξ 1 + |y|2

u( ˆ ξ ) dξ N(x, dy)

d

= −γ (x) · ∇u(x) + ∑ Qkl ∂kl u +

k,l

u(x) − u(x + y) +

Rd

y · ∇x u(x) 1 + |y|2

N(x, dy) .

AJC u

If we assume X to be an admissible market model, we can drop the damping factor 1 2 and replace the measure N by the corresponding jump kernel k in the above 1+|y|

calculation, due to (4.13). We convert the canonical jump operator AJC into the integrated jump operator AJ , due to an analogous argument to [93, Lemma 2.2.7], AJ u(x) =

d

∑

i=1 R d

+∑

(u(x + yiei ) − u(x) − yi∂i u(x)) ki (xi , yi )dyi

∑

Rj j=2 |I|= j I1 0 ⎪ ⎪ ⎪ ⎪ ⎪ 0.5 > x ≥ 0.25 ⎪ ⎨0.8x − 0.1, m(x) = 0.5 + k −0.4x + 0.5, 0.75 > x ≥ 0.5 . ⎪ ⎪ ⎪ ⎪−0.8x + 0.8, 1 > x ≥ 0.75 ⎪ ⎪ ⎪ ⎩0.5, else This process has no Gaussian component and the drift γ (x) is chosen according to (5.1).


187 Compressed stiffness matrix

Stiffness matrix 200

200

400

400

600

600

800

800

1000

1000

1200

1200

1400

1400

1600

1600

1800

1800 2000

2000

200 400 600 800 1000 1200 1400 1600 1800 2000

200 400 600 800 1000 1200 1400 1600 1800 2000

(a) Stiffness matrix

(b) Compressed stiffness matrix

Fig. 4 Stiffness matrices for the pure jump case with CGMY-type jump kernel 2 Y (x) = 1.25e−x + 0.5

50

50

100

100

150

150

200

200

250

250

300

300

350

350

400

400

450

450

500

500 50 100 150 200 250 300 350 400 450 500

(a) Stiffness matrix

50 100 150 200 250 300 350 400 450 500

(b) Mass matrix

Fig. 5 Stiffness and Mass matrices for the Black-Scholes model with σ = 0.3 and r = 0

In Figure 4 the stiffness matrix for the process in Example 9.1 is depicted. Note that the uncompressed stiffness matrix is densely populated, but structurally very similar to the matrix in the Black-Scholes model (Figure 5). In a next step we study the number of non-zero entries of the uncompressed and compressed stiffness matrix. Due to Section 6.4 we expect essentially linear growth of the number of non-zero elements for the compressed matrix (Figure 6). The condition numbers of the preconditioned stiffness matrices have to be uniformly bounded in the number of levels due to Section 7.1. A parameter study for various choices of k in Example 9.1 and Example 9.2 is shown in Figure 7. The condition numbers are uniformly bounded and of order 101 in most cases, although the norm equivalences (6.16) only apply to Example 9.1. For variable orders with

188

O. Reichmann and C. Schwab Compression

Number on non−zero entries

108 107

compressed full

106 105 104 s = 2.0 103 s = 1.0 102 101 100

101

102 103 Degrees of freedom

104

105

Fig. 6 Number of non-zero entries of the compressed/uncompressed stiffness matrix versus number of degrees of freedom corresponding to the Lévy kernel in Example 9.1 and k = 1.25 Parameter study for condition numbers

102

k=0.25 k=0.50 k=0.75 k=1.00 k=1.25 k=1.45 k=1.49

101

100 1 10


104

Condition number

Condition number

102

Parameter study for condition numbers k=4.00 k=2.00 k=1.50 k=1.25 k=1.00 k=0.75 k=0.50

101

100 1 10

(a) Example 9.1


104

(b) Example 9.2

Fig. 7 Condition numbers for different levels and choices of k

1.95 ≤ m we obtain condition numbers of order 102 . Note that the condition numbers are not only influenced by the order of the singularity of the jump kernel at z = 0, but also by the rates of exponential decay β + and β − . Fast decaying tails, i.e., large β + and β − may lead to larger constants. Figure 8 shows the price of a European put option for several Lévy processes and 2 one Feller process. In the Feller case we choose m(x) = 0.8e−x + 0.1 in Example 9.1 and for the Lévy models we set m ∈ {0.1, 0.5, 0.7, 0.8, 0.9}. In all cases we set C = 1, β + = β − = 10 and use truncation parameters a = −3, b = 3 in logmoneyness coordinates. The prices in the Feller model are significantly different from the prices in the different Lévy models. This can be explained by the ability of the Feller model to account for different tail behaviour for different states of the process, which is not possible using Lévy processes. Figure 9 shows the prices


189

50 Payoff Y=0.9 variable Y=0.8 Y=0.7 Y=0.5 Y=0.1

45 40

Option price

35 30 25 20 15 10 5 0 0.5

0.6

0.7

0.8

0.9 1 1.1 Moneyness

1.2

1.3

1.4

1.5

Fig. 8 Option prices for several models for a European put option with T = 1 and K = 100 60 Y=0.1 Y=0.5 Y=0.7 Y=0.8 Y=0.9 variable Payoff

50

Option price

40

30

20

10

0 0.5

0.6

0.7

0.8

0.9

1 1.1 Moneyness

1.2

1.3

1.4

1.5

Fig. 9 Option prices for several models for an American put option with T = 1 and K = 100

of American put option for a Feller process and a several Lévy models. We use a Lagrangian multiplier approach as described in Section 5.2 and refer to [48, 49] for more details, analogous results were obtained using the PSOR algorithm. The parameters were chosen as above.

190

O. Reichmann and C. Schwab 50 45

Option price Payoff Sensitivity wrt Y

40 Option price

35 30 25 20 15 10 5 0 0.5

0.6

0.7

0.8

0.9

1 1.1 Moneyness

1.2

1.3

1.4

1.5

Fig. 10 Computed sensitivity of a European put w.r. to the jump intensity parameter m in the CGMY model

We now consider model sensitivities. For the computation of sensitivities w.r. to model parameters we consider a special case of Example 9.1, i.e. k = 0 and therefore Y = 0.5 and calculate the sensitivity of the price w.r. to the jump intensity parameter m, where we let 0 < m < 2, the rest of the parameters being chosen as above. Then we have Sη = (0, 2) with η = m and A(δ m)ϕ = −δ m

*

+ ϕ (x + z) − ϕ (x) − z∂xϕ (x) k(z)dz ∈ L (V , V ∗ )

R

where the kernel k is given by k(z) := − ln |z|k(z). It is easy to check that it holds |z|≤1 z2 k(z)dz < ∞, |z|>1 k(z)dz < ∞ due to m < 2. 1 m/2+ ε (D) ⊂ H m/2 (D) = V , (D), if σ > 0, and V = H In this setting, V = V = H ∀ε > 0, if σ = 0. We refer to [44] for more details. Figure 10 shows the sensitivity in this model w.r. to the parameter η = m . As expected from Figure 8, we observe a positive sensitivity which is significantly larger at the money, than deep out or in the money.

9.2 Multidimensional Case We consider a special case of the model presented in Example 4.13 with constant model parameters and no diffusion, i.e., a multidimensional pure jump Lévy model. We are interested in option prices as well as the sensitivity with respect to the copula parameter ϑ . We have Sη = (0, ∞) with η = ϑ and A(δ ϑ ) = δ ϑ

R2

∂ 2u (x + y)Fϑ (U1 (y1 ),U2 (y2 )) dy, ∂ y1 ∂ y2

(9.1)


191

50 45 40 Option price

35 30 25 20 15 10 5 0 150

140

130

120

110

100

90

80 S1

70

60

50

150

140

130

120

100

110

90

80

70

60

50

S2

Fig. 11 European put basket price with payoff g(S1 , S2 ) = (K − 0.5S1 − 0.5S2 ) in a multidimensional CGMY model with Clayton copula with K = 100 and T = 1

2

Sensitivity

1.5 1 0.5 0 50 150 100

100 s1

150 50

s2

Fig. 12 Sensitivity with respect to the copula parameter ϑ of a European put basket price with payoff g(S1 , S2 ) = (K − 0.5S1 − 0.5S2 ) in a multidimensional CGMY model with Clayton copula with K = 100 and T = 1

192


where Fϑ is given by: d ϑ ∑di=1 |ui |−ϑ ln |ui | 1 −ϑ Fϑ (u) = 2 F(u) ln ∑ |ui | + . −ϑ ϑ ∑di=1 |ui | i=1 The following parameters were chosen C = 1, β − = [10, 9], β + = [15, 16], m = [0.5, 0.7], ϑ = 0.5, ρ = 0.5. The option price is depicted in Figure 11 and the sensitivity is depicted in Figure 12. Acknowledgements Partial Support by SNF under grant No. 144414 and by ERC under grant No. 247277 is gratefully acknowledged.

List of Symbols General Notation a∧b a∨b ab ab ab |I| Ic

min(a, b) max (a, b) There exists a positive constant K s.t. a ≤ Kb There exists a positive constant K s.t. a ≥ Kb a b and a b Cardinality of a finite set Complement of a set

Spaces of Functions and Distributions Bb (Rd ) C(Rd ) C∞ (Rd ) C0∞ (Rd ) H m (Rd ) H m (Rd ) H m(x) (Rd ) H m(x) (Rd ) m(x) (D) H S (Rd ) S ∗ (Rd )

Bounded Borel measurable functions on Rd Continuous functions on Rd Continuous functions on Rd vanishing at infinity Test functions on Rd Classical (fractional) Sobolev space on Rd Unisotropic (fractional) Sobolev space on Rd Isotropic variable order Sobolev space on Rd Unisotropic variable order Sobolev space on Rd Unisotropic variable order Sobolev space on an open Lipschitz domain D Schwartz space of tempered functions on Rd Tempered distributions


193

Functions and Operators a(x, D) D(A ) (Tt )t≥0 Ui (x, y) U I (x, y)

Pseudodifferential operator with symbol a(x, ξ ) Domain of an operator A One parameter semigroup of operators Tail integrals of a Feller process Multidimensional tail integrals of a Feller process

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Index to the contribution of Ken-iti Sato

ϒ -transformation, 43, 58

fractional integral, 6

absolutely continuous, 21, 56, 76 additive process, 3

Gaussian covariance matrix, 2 generalized Γ -convolutions, 11 Goldie-Steutel-Bondesson class, 10

class B, 10 class K∞,α , 9 class Kp,α , 9, 47 0 , 47 class Kp, α e , 47 class Kp, α class L, 2, 3 class L∞,α , 9 class L∞ , 5, 83 class L p,α , 9, 74 class L0p,α , 74 class Lep,α , 74 class T , 10 class U, 10 completely monotone, 7 completely selfdecomposable, 6 compound Poisson distribution, 56, 77 cumulant function, 2 domain absolute, 33 essential, 33 of Φ Lf , 36 of stochastic integral mapping, 32 elementary Γ -variable, 10 elementary compound Poisson variable, 10 elementary mixed-exponential variable, 10

improper stochastic integral, 3, 32 absolutely definable, 33 definable, 32 essentially definable, 33 infinitely divisible distribution, 2 Jurek class, 10 Lévy measure, 2 Lévy process, 3 Lévy–Khintchine triplet, 26 Lamperti transformation, 4 locally finite, 7 locally square-integrable, 31 location parameter, 27 log-normal distribution, 11 lower semi-continuous, 20 mapping Φ f , 8, 32 measurable family, 21 mixture of exponential distributions, 11 monotone of order p, 7, 12 multiply selfdecomposable, 4 n times selfdecomposable, 4, 75 fractional times selfdecomposable, 4 twice selfdecomposable, 4

197

198 nondegenerate, 56 null array, 3

of polar product type, 29 Ornstein–Uhlenbeck process driven by Lévy process, 3 Ornstein–Uhlenbeck type process, 3

Pareto distribution, 11 polar decomposition, 2, 27

radial decomposition, 2, 27 range absolute, 33 essential, 33 of Φ Lf , 37 of stochastic integral mapping, 33 Riemann–Liouville integral, 6

Index selfdecomposable, 2 selfsimilar, 3 semigroup property, 7 spherical decomposition, 28 stable distribution, 5, 65 strictly, 5, 65 stochastic area, 11 stochastic integral, 32

tempered stable distribution, 61 Thorin class, 10 transformation Φ Lf , 36 triplet, 26

vague convergence, 17, 82

weak mean, 29, 49 have weak mean absolutely, 30, 52

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Recent Reprints and New Editions Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999 – Corr. 3rd printing (2007) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 – 2nd corr. and augmented edition (2007) Vol. 1693: S. Simons, From Hahn-Banach to Monotonicity (Minimax and Monotonicity 1998) – 2nd exp. edition (2008) Vol. 470: R.E. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. With a preface by D. Ruelle. Edited by J.-R. Chazottes. 1975 – 2nd rev. edition (2008) Vol. 523: S.A. Albeverio, R.J. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integral. 1976 – 2nd corr. and enlarged edition (2008) Vol. 1764: A. Cannas da Silva, Lectures on Symplectic Geometry 2001 – Corr. 2nd printing (2008)

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Levy matters I: Foundations, trees and numerical issues in finance

Numerical methods in finance

Numerical techniques in finance

Issues in Finance and Monetary Policy

Numerical Methods for Finance

Numerical Methods for Finance

Levy Processes in Credit Risk (The Wiley Finance Series)

Numerical Grid Generation: Foundations and Applications

Fractals in engineering Levy Vehel

Key Issues In Bilingualism and Bilingual Education (Multilingual Matters 35)

Valuation Approaches and Metrics (Foundations and Trends in Finance)

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