Stochastic Processes and Applications to Mathematical Finance: Proceedings of the 6th International Symposium, Ritsumeikan University, Japan, 6-10 March 2006

Proceedings of the 6th Ritsumelkan International Symposium STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINAN...

Author: Jiro Akahori | Shigeyoshi Ogawa | Shinzo Watanabe

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Proceedings of the 6th Ritsumelkan International Symposium

STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE

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Proceedings of the 6the Ritsumeikan International Symposium

STOCHASTIC PROCESSES AND APPLICATIONS TO

MATHEMATICAL FINANCE Ritsumeikan University,, Japan

6–10 March 2006

Editors

Joro Akahori Shigeyoshi Ogawa Shinzo Watanabe Ritsumeikan University,, Japan

World Scientific NEW JERSEY . LONDON . SINGAPORE . BEIJING . SHANGHAI . HONG KONG . TAIPEI . CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Proceedings of the 6th Ritsumeikan International Symposium Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-270-413-9 ISBN-10 981-270-413-2

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PREFACE The 6th Ritsumeikan international conference on Stochastic Processes and Applications to Mathematical Finance was held at Biwako-Kusatsu Campus (BKC) of Ritsumeikan University, March 6–10, 2006. The conference was organized under the joint auspices of Research Center for Finance and Department of Mathematical Sciences of Ritsumeikan University, and financially supported by MEXT (Ministry of Education, Culture, Sports, Science and Technology) of Japan, the Research Organization of Social Sciences, Ritsumeikan University, and Department of Mathematical Sciences, Ritsumeikan University. The series of the Ritsumeikan conferences has been aimed to hold assemblies of those interested in the applications of theory of stochastic processes and stochastic analysis to financial problems. The Conference, counted as the 6th one, was also organized in this line: there several eminent specialists as well as active young researchers were jointly invited to give their lectures (see the program cited below) and as a whole we had about hundred participants. The present volume is the proceedings of this conference based on those invited lectures. We, members of the editorial committee listed below, would express our deep gratitude to those who contributed their works in this proceedings and to those who kindly helped us in refereeing them. We would express our cordial thanks to Professors Toshio Yamada, Keisuke Hara and Kenji Yasutomi at the Department of Mathematical Sciences, of Ritsumeikan University, for their kind assistance in our editing this volume. We would thank also Mr. Satoshi Kanai for his works in editing TeX files and Ms. Chelsea Chin of World Scientific Publishing Co. for her kind and generous assistance in publishing this proceedings. December, 2006, Ritsumeikan University (BKC) Jiroˆ Akahori Shigeyoshi Ogawa Shinzo Watanabe

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The 6th Ritsumeikan International Conference on STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Date March 6–10, 2006 Place Rohm Memorial Hall/Epoch21, in BKC, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan Program March, 6 (Monday): at Rohm Memorial Hall 10:00–10:10 Opening Speech, by Shigeyoshi Ogawa (Ritsumeikan University) 10:10–11:00 T. Lyons (Oxford University) Recombination and cubature on Wiener space 11:10–12:00 S. Ninomiya (Tokyo Institute of Technology) Kusuoka approximation and its application to finance 12:00–13:30 Lunch time 13:30–14:20 T. Fujita (Hitotsubashi University, Tokyo) Some results of local time, excursion in random walk and Brownian motion 14:30–15:20 K. Hara (Ritsumeikan University, Shiga) Smooth rough paths and the applications 15:20–15:50 Break 15:50–16:40 X-Y Zhou (Chinese University of Hong-Kong) Behavioral portfolio selection in continuous time 17:30– Welcome party March, 7 (Tuesday): at Rohm Memorial Hall 10:00–10:50 M. Schweizer (ETH, Zurich) Aspects of large investor models 11:10–12:00 J. Imai (Tohoku University, Sendai) A numerical approach for real option values and equilibrium strategies in duopoly 12:00–13:30 Lunch time


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13:30–14:20 H. Pham (Univ. Paris VII) An optimal consumption model with random trading times and liquidity risk and its coupled system of integrodifferential equations 14:30–15:20 K. Hori (Ritsumeikan University, Shiga) Promoting competition with open access under uncertainty 15:20–15:50 Break 15:50–16:40 K. Nishioka (Chuo University, Tokyo) Stochastic growth models of an isolated economy March, 8 (Wednesday): at Rohm Memorial Hall 10:00–10:50 H. Kunita (Nanzan University, Nagoya) Perpetual game options for jump diffusion processes 11:10–11:50 E. Gobet (Univ. Grenoble) A robust Monte Carlo approach for the simulation of generalized backward stochastic differential equations 12:00– Excursion March, 9 (Thursday): at Epoch21 10:00–10:50 P. Imkeller (Humbold University, Berlin) Financial markets with asymmetric information: utility and entropy 11:00–12:00 M. Pontier (Univ. Toulouse III) Risky debt and optimal coupon policy 12:00–13:30 Lunch time 13:30–14:20 H. Nagai (Osaka University) Risk-sensitive quasi-variational inequalities for optimal investment with general transaction costs 14:30–15:20 W. Runggaldier (Univ. Padova) On filtering in a model for credit risk 15:20–15:50 Break 15:50–16:40 D. A. To (Univ. Natural Sciences, HCM city) A mixed-stable process and applications to option pricing 16:50– Short Communications 1. Y. Miyahara (Nagoya City University) 2. T. Tsuchiya (Ritsumeikan University, Shiga) 3. K. Yasutomi (Ritsumeikan University, Shiga)


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March, 10 (Friday): Epoch21 10:00–10:50 R. Cont (Ecole Polytechnique, France) Parameter selection in option pricing models: a statistical approach 11:10–12:00 T. V. Nguyen (Hanoi Institute of Mathematics) Multivariate Bessel processes and stochastic integrals 12:00–13:30 Lunch time 13:30–14:20 J-A, Yan (Academia Sinica, China) A functional approach to interest rate modelling 14:30–15:20 M. Arisawa (Tohoku University, Sendai) A localization of the Lévy operators arising in mathematical finances 15:20–15:50 Break 15:50–16:40 A. N. Shiryaev (Steklov Mathem. Institute, Moscow) Some explicit stochastic integral representation for Brownian functionals 18:30– Reception at Kusatsu Estopia Hotel


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contents

CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Financial Markets with Asymmetric Information: Information Drift, Additional Utility and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Ankirchner and P. Imkeller

1

A Localization of the Lévy Operators Arising in Mathematical Finances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Arisawa

23

Model-free Representation of Pricing Rules as Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Biagini and R. Cont

53

A Class of Financial Products and Models Where Super-replication Prices are Explicit . . . . . . . L. Carassus, E. Gobet, and E. Temam

67

Risky Debt and Optimal Coupon Policy and Other Optimal Strategies . . . . . . . . . . . . . . . . . . . . . . D. Dorobantu and M. Pontier

85

Affine Credit Risk Models under Incomplete Information . . . . . . . . . . . . . . . . . R. Frey, C. Prosdocimi, and W. J. Runggaldier

97

Smooth Rough Paths and the Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Hara and T. Lyons

115

From Access to Bypass: A Real Options Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Hori and K. Mizuno

127

The Investment Game under Uncertainty: An Analysis of Equilibrium Values in the Presence of First or Second Mover Advantage. . . . . . . . . . . . . . . . . . . . . . . . . . . J. Imai and T. Watanabe

151

Asian Strike Options of American Type and Game Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Ishihara and H. Kunita

173

Minimal Variance Martingale Measures for Geometric Lévy Processes . . . . . . . . . M. Jeanblanc, S. Kloeppel, and Y. Miyahara

193

xi

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contents

xii

Cubature on Wiener Space Continued . . . . C. Litterer and T. Lyons

197

A Remark on Impulse Control Problems with Risk-sensitive Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Nagai

219

A Convolution Approach to Multivariate Bessel Proceses . . . . . . . . . . . . . . . . . . . . T. V. Nguyen, S. Ogawa, and M. Yamazato

233

Spectral Representation of Multiply Self-decomposable Stochastic Processes and Applications . . . . . . . . . . . . . N. V. Thu, T. A. Dung, D. T. Dam, and N. H. Thai

245

Stochastic Growth Models of an Isolated Economy . . . K. Nishioka

259

Numerical Approximation by Quantization for Optimization Problems in Finance under Partial Observations . . . . H. Pham

275

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01Ankirchner˙Imkeller

Financial Markets with Asymmetric Information: Information Drift, Additional Utility and Entropy Stefan Ankirchner and Peter Imkeller Institut fur ¨ Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

We review a general mathematical link between utility and information theory appearing in a simple financial market model with two kinds of small investors: insiders, whose extra information is stored in an enlargement of the less informed agents’ filtration. The insider’s expected logarithmic utility increment is described in terms of the information drift, i.e. the drift one has to eliminate in order to perceive the price dynamics as a martingale from his perspective. We describe the information drift in a very general setting by natural quantities expressing the conditional laws of the better informed view of the world. This on the other hand allows to identify the additional utility by entropy related quantities known from information theory. Key words: enlargement of filtration; logarithmic utility; utility maximization; heterogeneous information; insider model; Shannon information; information difference; entropy. 2000 AMS subject classifications: primary 60H30, 94A17; secondary 91B16, 60G44. 1. Introduction A simple mathematical model of two small agents on a financial market one of which is better informed than the other has attracted much attention in recent years. Their information is modelled by two different filtrations: the less informed agent has the σ−field Ft , corresponding to the natural evolution of the market up to time t at his disposal, while the better informed insider knows the bigger σ−field Gt ⊃ Ft . Here is a short selection of some among many more papers dealing with this model. Investigation techniques concentrate on martingale and stochastic control theory, and methods of enlargement of filtrations (see Yor , Jeulin , Jacod in [22]), starting with the conceptual paper by Duffie, Huang [12]. The model 1

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2

is successively studied on stochastic bases with increasing complexity: e.g. Karatzas, Pikovsky [24] on Wiener space, Grorud, Pontier [15] allow Poissonian noise, Biagini and Oksendal [7] employ anticipative calculus techniques. In the same setting, Amendinger, Becherer and Schweizer [1] calculate the value of insider information from the perspective of specific utilities. Baudoin [6] introduces the concept of weak additional information, while Campi [8] considers hedging techniques for insiders in the incomplete market setting. Many of the quoted papers deal with the calculation of the better informed agent’s additional utility. In Amendinger et al. [2], in the setting of initial enlargements, the additional expected logarithmic utility is linked to information theoretic concepts. It is computed in terms of an energy-type integral of the information drift between the filtrations (see [18]), and subsequently identified with the Shannon entropy of the additional information. Also for initial enlargements, Gasbarra, Valkeila [14] extend this link to the Kullback-Leibler information of the insider’s additional knowledge from the perspective of Bayesian modelling. In the environment of this utility-information paradigm the papers [16], [19], [17], [18], Corcuera et al. [9], and Ankirchner et al. [5] describe additional utility, treat arbitrage questions and their interpretation in information theoretic terms in increasingly complex models of the same base structure. Utility concepts different from the logarithmic one correspond on the information theoretic side to the generalized entropy concepts of f −divergences. In this paper we review the main results about the interpretation of the better informed trader’s additional utility in information theoretic terms mainly developed in [4], concentrating on the logarithmic case. This leads to very basic problems of stochastic calculus in a very general setting of enlargements of filtrations: to ensure the existence of regular conditional probabilities of σ–fields of the larger with respect to those of the smaller filtration, we only eventually assume that the base space be standard Borel. In Section 2, we calculate the logarithmic utility increment in terms of the information drift process. Section 3 is devoted to the calculation of the information drift process by the Radon-Nikodym densities of the stochastic kernel in an integral representation of the conditional probability process and the conditional probability process itself. For convenience, before proceeding to the more abstract setting of a general enlargement, the results are given in the initial enlargement framework first. In Section 4 we finally provide the identification of the utility increment in the general enlargement setting with the information difference of the two filtrations in terms of Shannon entropy concepts.

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2. Additional Logarithmic Utility and Information Drift Let us first fix notations for our simple financial market model. First of all, to simplify the exposition, we assume that the trading horizon is given by T = 1. Let (Ω, F , P) be a probability space with a filtration (Ft )0≤t≤1 . We consider a financial market with one non-risky asset of interest rate normalized to 0, and one risky asset with price Xt at time t ∈ [0, 1]. We assume that X is a continuous (Ft )−semimartingale with values in R and write A for the set of all X−integrable and (Ft )−predictable processes θ such that θ0 = 0. If θ ∈ A, then we denote by (θ · S) the usual stochastic integral process. For all x > 0 we interpret x + (θ · X)t , 0 ≤ t ≤ 1, as the wealth process of a trader possessing an initial wealth x and choosing the investment strategy θ on the basis of his knowledge horizon corresponding to the filtration (Ft ). Throughout this paper we will suppose the preferences of the agents to be described by the logarithmic utility function. Therefore it is natural to suppose that the traders’ total wealth has always to be strictly positive, i.e. for all t ∈ [0, 1] (1)

Vt (x) = x + (θ · X)t > 0 a.s.

Strategies θ satisfying Eq. (1) will be called x−superadmissible. The agents want to maximize their expected logarithmic utility from terminal wealth. So we are interested in the exact value of u(x) = sup{E log(V1 (x)) : θ ∈ A, x − superadmissible}. Sometimes we will write uF (x), in order to stress the underlying filtration. The expected logarithmic utility of the agent can be calculated easily, if one has a semimartingale decomposition of the form (2)

t

Xt = Mt +

ηs dM, Ms , 0

where η is a predictable process. Such a decomposition has to be expected in a market in which the agent trading on the knowledge flow (Ft ) has no arbitrage opportunities. In fact, if X satisfies the property (NFLVR), then it may be decomposed as in Eq. (2) (see [10]). It is shown in [3] that finiteness of u(x) already implies the validity of such a decomposition. Hence a decomposition as in (2) may be given even in cases where arbitrage exists. We state Theorem 2.9 of [5], in which the basic relationship between optimal logarithmic utility and information related quantities becomes visible.

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Proposition 2.1. Suppose X can be decomposed into X = M + η · M, M. Then for any x > 0 the following equation holds (3)

u(x) = log(x) +

1 E 2

1

0

η2s dM, Ms .

Let us give the core arguments proving this statement in a particular setting, and for initial wealth x = 1. Suppose that X is given by the linear sde dXt = αt dt + dWt , Xt with a one-dimensional Wiener process W, and assume that the small trader’s filtration (Ft ) is the (augmented) natural filtration of W. Here α is a progressively measurable mean rate of return process which satisfies 1 |αt |dt < ∞, P−a.s. Let us denote investment strategies per unit by π, so 0 that the wealth process V(x) is given by the simple linear sde dVt (x) dXt = πt · . Vt (x) Xt It is obviously solved by the formula

t

Vt (x) = exp[

πs dWs − 0

1 2

t 0

π2s ds +

t

πs αs ds]. 0

t Due to the local martingale property of 0 πs dWs , t ∈ [0, 1], the expected logarithmic utility of the regular trader is deduced from the maximization problem (4)

uF (1) = max E[ π

1

πs αs ds − 0

1 2

1 0

π2s ds].

The maximization of

1

πs αs ds −

π → 0

1 2

1 0

π2s ds

for given processes α is just a more complex version of the one-dimensional maximization problem for the function 1 π → π α − π2 2

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5

with α ∈ R. Its solution is obtained by the critical value π = α and thus 1 1 α2s ds]. (5) uF (1) = E[ 2 0 This confirms the claim of Proposition 2.1. This proposition motivates the following definition. Definition 2.1. A filtration (G t ) is called finite utility filtration for X, if X is a (Gt )−semimartingale with decomposition dX = dM + ζ · dM, M, where 1 ζ is (Gt )−predictable and belongs to L2 (M), i.e. E 0 ζ2 dM, M < ∞. We write F = {(Ht ) ⊃ (Ft )(Ht ) is a finite utility filtration for X}. We now compare two traders who take their portfolio decisions not on the basis of the same filtration, but on the basis of different information flows represented by the filtrations (Gt ) and (Ht ) respectively. Suppose that both filtrations (Gt ) and (Ht ) are finite utility filtrations. We denote by (6)

X = M + ζ · M, M

the semimartingale decomposition with respect to (Gt ) and by (7)

X = N + β · N, N

the decomposition with respect to (Ht ). Obviously, M, M = X, X = N, N and therefore the utility difference is equal to 1 1 uH (x) − uG (x) = E (β2 − ζ2 ) dM, M. 2 0 Furthermore, Eqs. (6) and (7) imply (8)

M = N − (ζ − β) · M, M

a.s.

If Gt ⊂ Ht for all t ≥ 0, Eq. (8) can be interpreted as the semimartingale decomposition of M with respect to (Ht ). In this case one can show that the utility difference depends only on the process µ = ζ − β. In fact, 1 1 (β2 − ζ2 ) dM, M uH (x) − uG (x) = E 2 0 1 1 1 2 µ dM, M) − E( µ ζ dM, M) = E( 2 0 0 1 1 = E( µ2 dM, M). 2 0

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The last equation is due to the fact that N − M = µ dM, M is a martingale with respect to (Ht ), and ζ is adapted to this filtration. It is therefore natural to relate µ to a transfer of information. Definition 2.2. Let (G t ) be a finite utility filtration and X = M + ζ · M, M the Doob-Meyer decomposition of X with respect to (Gt ). Suppose that (Ht ) is a filtration such that Gt ⊂ Ht for all t ∈ [0, 1]. The (H t )−predictable process µ satisfying

·

M−

µt dM, Mt

is a (Ht ) − local martingale

0

is called information drift (see [18]) of (H t ) with respect to (Gt ). The following proposition summarizes the findings just explained, and relates the information drift to the expected logarithmic utility increment. Proposition 2.2. Let (G t ) and (Ht ) be two finite utility filtrations such that Gt ⊂ Ht for all t ∈ [0, 1]. If µ is the information drift of (H t ) w.r.t. (Gt ), then we have 1 1 µ2 dM, M. uH (x) − uG (x) = E 2 0 3. The Information Drift and the Law of Additional Information In this section we aim at giving a description of the information drift between two filtrations in terms of the laws of the information increment between two filtrations. This is done in two steps. First, we shall consider the simplest possible enlargement of filtrations, the well known initial enlargement. In a second step, we shall generalize the results available in the initial enlargement framework. In fact, we consider general pairs of filtrations, and only require the state space to be standard Borel in order to have conditional probabilities available. 3.1 Initial enlargement, Jacod’s condition In this setting, the additional information in the larger filtrations is at all times during the trading interval given by the knowledge of a random variable which, from the perspective of the smaller filtration, is known only at the end of the trading interval. To establish the concepts in fair simplicity, we again assume that the smaller underlying filtration (Ft ) is the augmented filtration of a one-dimensional Wiener process W. Let G be an F1 –measurable random variable, and let Gt = Ft ∨ σ(G),

t ∈ [0, 1].

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Suppose that (Gt ) is small enough so that W is still a semimartingale with respect to this filtration. More precisely, suppose that there is an information drift µG such that 1 |µG s | ds < ∞ P-a.s., 0

and such that

.

˜ + W=W

(9)

0

µG s ds

˜ To clarify the relationship between the with a (Gt )− Brownian motion W. additional information G and the information drift µG , we shall work under a condition concerning the laws of the additional information G which has been used as a standing assumption in many papers dealing with grossissement de filtrations. See Yor [27], [26], [28], Jeulin [21]. The condition was essentially used in the seminal paper by Jacod [20], and in several equivalent forms in Follmer ¨ and Imkeller [13]. To state and exploit it, let us first mention that all stochastic quantities appearing in the sequel, often depending on several parameters, can always be shown to possess measurable versions in all variables, and progressively measurable versions in the time parameter (see Jacod [20]). Denote by PG the law of G, and for t ∈ [0, 1], ω ∈ Ω, by P G t (ω, dl) the regular conditional law of G given Ft at ω ∈ Ω. Then the condition, which we will call Jacod’s condition, states that (10) PGt (ω, dg) is absolutely continuous with respect to PG (dg) for P− a.e. ω ∈ Ω.

Also its reinforcement (11)

PG t (ω, dg) is equivalent to

PG (dg) for P− a.e. ω ∈ Ω,

will be of relevance. Denote the Radon-Nikodym density process of the conditional laws with respect to the law by pt (ω, g) =

dPG t (ω, ·) dPG

(g),

g ∈ R, ω ∈ Ω.

By the very definition, t → Pt (·, dg) is a local martingale with values in the space of probability measures on the Borel sets of R. This is inherited to t → pt (·, g) for (almost) all g ∈ R. Let the representations of these martingales with respect to the (Ft )−Wiener process W be given by t g ku dWu , t ∈ [0, 1] pt (·, g) = p0 (·, g) + 0

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with measurable kernels k. To calculate the information drift in terms of these kernels, take s, t ∈ [0, 1], s ≤ t, and let A ∈ Fs and a Borel set B on the real line determine the typical set A ∩ G−1 [B] in a generator of Gs . Then we may write E([Wt − Ws ] 1A 1B (G)) = E( 1A [Wt − Ws ] PG t (·, dg)) B = E(1A [Wt − Ws ] [pt − ps ](·, g)) PG(dg)

B

=

t

g

ku du) PG (dg)

E(1A

B

=

s

s

g

t

ku pu (·, g) du) PG(dg) pu (·, g)

t

ku du pt (·, g)) PG(dg) pu (·, g)

E(1A

B

=

g

E(1A s

B

g

ku PG (·, dg)) p (·, g) t u B t g ku | g=G du). = E(1A 1B (G) s pu (·, g) = E(

1A

The bottom line of this chain of arguments shows that · klu ˜ =W− | g=G du W 0 pu (·, g) is a (G )−martingale, hence a (Gt )−Brownian motion provided that 1 kg t | u | | du < ∞ P−a.s.. This completes the deduction of an explicit 0 pu (·,g) g=G formula for the information drift of G in terms of quantities related to the law of G in which we use the common oblique bracket notation to denote the covariation of two martingales (for more details see Jacod [20]). Theorem 3.1. Suppose that Jacod’s condition (10) is satisfied, and furthermore that g

(12)

µG t

kt | g=G = = pt (·, g)

d dt p(·,

g), Wt

pt (·, g)

| g=G ,

satisfies

1

(13) 0

|µG u | du < ∞ P−a.s..

t ∈ [0, 1],

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9

Then

·

˜ + W=W 0

µG s ds

˜ is a G−semimartingale with a G−Brownian motion W. To see how restrictive condition (10) may be, let us illustrate it by looking at two possible additional information variables G. Example 1: Let > 0 and suppose that the stock price process is a regular diffusion given by a stochastic differential equation with bounded volatility σ and drift α, σt = σ(Xt ), t ∈ [0, 1], where σ is a smooth function without zeroes. Let G = X1+ . Then in particular X is a time homogeneous Markov process with transition probabilities Pt (x, dy), x ∈ R+ , t ∈ [0, 1], which are equivalent with Lebesgue measure on R+ . For t ∈ [0, 1], the regular conditional law of G given Ft is then given by P1+ −t (Xt , dy), which is equivalent with the law of G. Hence in this case, even the strong version of Jacod’s hypothesis (11) is verified. Example 2: Let G = sup Wt . t∈[0,1]

To abbreviate, denote for t ∈ [0, 1] Gt = sup Ws , 0≤s≤t

G˜ 1−t = sup (Ws − Wt ). t≤s≤1

Finally, let p1−t denote the density function of G˜ 1−t . Then we may write for every t ∈ [0, 1] G = Gt ∨ [Wt + G˜ 1−t ].

(14)

Now Gt is Ft −measurable, independent of G˜ 1−t , and therefore for Borel sets A on the real line we have Gt −Wt (·, A) = p (y)dy · δ (A) + p1−t (y)dy. (15) PG 1−t G t t −∞

A∩[Gt −Wt ,∞[

Note now that the family of Dirac measures in the first term of (15) is supported on the random points Gt , and that the law of Gt is absolutely continuous with respect to Lebesgue measure on R+ . Hence there cannot be any common reference measure equivalent with δGt P−a.s. Therefore in this example Jacod’s condition is violated.

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10

It can be seen that there is an extension of Jacod’s framework into which example 2 still fits. This is explained in [18], [19], and resides on a version of Malliavin’s calculus for measure valued random elements. It yields a description of the information drift in terms of traces of logarithmic Malliavin gradients of conditional laws of G. We shall not give details here, since we will go a considerable step ahead of this setting. In fact, in the following subsection we shall further generalize the framework beyond the Wiener space setting. 3.2 General enlargement Assume again that the price process X is a semimartingale of the form X = M + η · M, M with respect to a finite utility filtration (Ft ). Moreover, let (Gt ) be a filtration such that Ft ⊂ Gt , and let α be the information drift of (Gt ) relative to (Ft ). We shall explain how the description of α by basic quantities related to the conditional probabilities of the larger σ−algebras Gt with respect to the smaller ones Ft , t ≥ 0 generalizes from the setting of the previous subsection. Roughly, the relationship is as follows. Suppose for all t ≥ 0 there is a regular conditional probability Pt (·, ·) of F given Ft , which can be decomposed into a martingale component orthogonal to M, plus a component possessing a stochastic integral representation with respect to M with a kernel function kt (·, ·). Then, provided α is square integrable with respect to dM, M ⊗ P, the kernel function at t will be a signed measure in its set variable. This measure is absolutely continuous with respect to the conditional probability itself, if restricted to Gt , and α coincides with their Radon-Nikodym density. As a remarkable fact, this relationship also makes sense in the reverse direction. Roughly, if absolute continuity of the stochastic integral kernel with respect to the conditional probabilities holds, and the RadonNikodym density is square integrable, the latter turns out to provide an information drift α in a Doob-Meyer decomposition of X in the larger filtration. To provide some details of this fundamental relationship, we need to work with conditional probabilities. We therefore assume that (Ω, F , P) is standard Borel (see [23]). Unfortunately, since we have to apply standard techniques of stochastic analysis, the underlying filtrations have to be assumed completed as a rule. On the other hand, for handling conditional probabilities it is important to have countably generated conditioning σ– fields. For this reason we shall use small versions (Ft0 ), (G0t ) which are countably generated, and big versions (Ft ), (Gt ) that are obtained as the smallest right-continuous and completed filtrations containing the small

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11

ones, and thus satisfy the usual conditions of stochastic calculus. We further suppose that F0 is trivial and that every (Ft )−local martingale has a continuous modification, and of course Ft0 ⊂ G0t for all t ≥ 0. We assume that M a (Ft0 )−local martingale. The regular conditional probabilities relative to the σ−algebras Ft0 are denoted by Pt . For any set A ∈ F the process (t, ω) → Pt (ω, A) is an (Ft0 )−martingale with a continuous modification adapted to (Ft ) (see e.g. Theorem 4, Chapter VI in [11]). We may assume that the processes Pt (·, A) are modified in such a way that Pt (ω, ·) is a measure on F for PM −almost all (ω, t), where PM is given on Ω × [0, 1] defined by PM (Γ) = ∞ E 0 1Γ (ω, t)dM, Mt , Γ ∈ F ⊗B+ . It is known that each of these martingales may be described in the unique representation (see e.g. [25], Chapter V) t (16) Pt (·, A) = P(A) + ks (·, A)dMs + LA t , 0

where k(·, A) is (Ft )−predictable and LA satisfies LA , M = 0. Note that trivially each σ−field in the left-continuous filtration (G0t− ) is also generated by a countable number of sets. We claim that the existence of an information drift of (Gt ) relative to (Ft ) for the process M depends on the validity of the following condition, which is the generalization of Jacod’s condition (10) to arbitrary stochastic bases on standard Borel spaces. Condition 3.1. k t (ω, ·)G0 is a signed measure and satisfies t−

kt (ω, ·)

G0t−

Pt (ω, ·)

G0t−

for PM −a.a (ω, t). If (3.1) is satisfied, one can show (see [4]) that there exists an (F t ⊗ Gt )−predictable process γ such that for PM −a.a. (ω, t) dkt (ω, ·) (ω ). (17) γt (ω, ω ) = dPt (ω, ·) G0t− It is also immediate from the definition that (18)

γt (ω, ω ) Pt (ω, dω ) dM, Mt = γt (ω, ω) dM, Mt .

On the basis of these simple facts it is possible to identify the information drift, provided (3.1) is guaranteed.

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12

Theorem 3.1. Suppose Condition 3.1 is satisfied and γ is as in (17). Then αt (ω) = γt (ω, ω) is the information drift of (Gt ) relative to (Ft ). Proof. We give the arguments in case M is a martingale. For 0 ≤ s < t and A ∈ G0s we have to show t E [1A (Mt − Ms )] = E 1A γu (ω, ω) dM, Mu . s

Observe E [1A (Mt − Ms )] = E [Pt (·, A)(Mt − Ms )] t = E (Mt − Ms ) ku (·, A) dMu + E[(Mt − Ms )LA t ] 0

t

=E

ku (·, A) dM, Mu s

t

γu (ω, ω ) dPu (ω, dω ) dM, Mu

=E s

A

t = E 1A (ω) γu (ω, ω) dM, Mu , s

where we used (18) in the last equation. We now look at the problem from the reverse direction. As an immediate consequence of (18) and Proposition 2.2 note that (Gt ) is a finite utility filtration if and only if γ2t (ω, ω ) Pt (ω, dω ) dM, Mt dP(ω) < ∞. Starting with the assumption that (Gt ) is a finite utility filtration, which 1 thus amounts to E 0 α2 dM, M < ∞, we derive the validity of Condition 3.1. In the sequel, (Gt ) denotes a finite utility filtration and α its predictable information drift, i.e. · ˜ αt dM, Mt (19) M=M− 0

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13

is a (Gt )−local martingale. To prove absolute continuity, we first define approximate Radon-Nikodym densities. This will be done along a sequence of partitions of the state space which generate the respective σ–fields of the bigger filtration. So let tni = 2in for all n ≥ 0 and 0 ≤ i ≤ 2n . We denote by T the set of all tni . It is possible to choose a family of finite partitions (Pi,n ) such that • for all t ∈ T we have G0t− = σ(Pi,n : i, n ≥ 0 s.t. tni = t), • Pi,n ⊂ Pi+1,n , • if i < j, n < m and i 2−n = j 2−m , then Pi,n ⊂ P j,m . We define for all n ≥ 0 the following approximate Radon-Nikodym densities γnt (ω, ω )

=

n 2 −1

i=0 A∈Pi,n

1]tni ,tni+1 ] (t)1A (ω )

kt (ω, A) . Pt (ω, A)

k (ω,A)

Note that Ptt (ω,A) is (Ft )−predictable and 1 ]tni ,tni+1 ] (t)1A(ω ) is (Gt )−predictable. Hence the product of both functions, defined as a function on Ω2 × [0, 1], is predictable with respect to (Ft ⊗ Gt ). By the very definition, for PM −almost all (ω, t) ∈ Ω×[0, 1] the discrete process (γ m t (ω, ·))m≥1 is a martingale. To have a chance to see this martingale converge as m → ∞, we will prove uniform integrability which will follow from the boundedness of the sequence in L2 (Pt (ω, ·)). This again is a consequence of the following key inequality (for more details see [4]). Lemma 3.1. Let 0 ≤ s < t ≤ 1 and P = {A 1 , . . . , An } be a finite partition of Ω into G0s −measurable sets. Then 2 t t n ku 2 E (·, Ak ) 1Ak dM, Mu ≤ 4E αu dM, Mu < ∞. Pu s s k=1

Proof. An application of Ito’s formula, in conjunction with (16) and (19),

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14

yields n

1Ak log Ps (·, Ak ) − 1Ak log Pt (·, Ak ) k=1

=

n k=1

t

−

s

1 1A dPu (·, Ak ) Pu (·, Ak ) k

1 1 dP(·, A ), P(·, A ) k k u 2 Ak s Pu (·, Ak ) t t ku ku ˜u− (·, Ak ) 1Ak dM (·, Ak ) 1Ak αu dM, Mu − = s Pu s Pu k=1 t 2 1 1 t ku Ak 1Ak dLu + − (·, Ak ) 1Ak dM, Mu 2 s Pu s Pu (·, Ak ) 1 t 1 Ak Ak + 1 dL , L A u 2 s Pu (·, Ak )2 k 1 2 n

t

+

(20)

Note that Pt (·, Ak ) log Pt (·, Ak ) is a submartingale bounded from below for all k. Hence the expectation of the left hand side in the previous equation is at most 0. One readily sees that the stochastic integral process with respect ˜ in this expression is a martingale and hence has vanishing expectation, to M while a similar statement holds for the stochastic integral with respect to the singular parts LAk . Consequently we may deduce from Eq. (20) and the Kunita-Watanabe inequality 2 n 1 t ku (·, Ak ) 1Ak dM, Mu E 2 s Pu k=1 n t ku ≤E (·, Ak ) 1Ak αu dM, Mu s Pu k=1 ⎞ 12 ⎛ n 2 12 t ⎟⎟ ⎜⎜ t ku 2 ⎟ ⎜ (·, Ak ) 1Ak dM, Mu ⎟⎟⎠ E αu dM, Mu , ≤ E ⎜⎜⎝ Pu s s k=1

which implies E

2 t t n ku (·, Ak ) 1Ak dM, Mu ≤ 4E α2u dM, Mu . Pu s s k=1

This completes the proof.

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15

Lemma 3.1 will now allow us to obtain a Radon-Nikodym density 1 process provided the given information drift α satisfies E 0 α2 dM, M < ∞. Note that our main result implicitly contains the statement that the kernel kt is a signed measure on the σ–field G0t , PM −a.e. 1 Theorem 3.2. Suppose that the information drift α satisfies E 0 α2 dM, M < ∞. Then the kernel k is absolutely continuous with respect to Pt (ω, ·)|G0t− , for PM −a.a. (ω, t) ∈ Ω × [0, 1]. This means that Condition 3.1 is satisfied. Moreover, the density process γ provides a description of the information drift of (G t ) relative to (Ft ) by the formula αt (ω) = γt (ω, ω). 2 Proof. By definition and Lemma 3.1 (γm t (ω, ·))m≥1 is an L (Pt (ω, ·))–bounded m martingale and hence, for a.a. fixed (ω, t), (γ t (ω, ·))m≥1 possesses a limit γ. It can be chosen to be (Ft ⊗ Gt )−predictable. Take for example

γt = lim inf(γnt ∨ 0) + lim sup(γnt ∧ 0). n

n

Now define a signed measure by ˜kt (ω, A) = 1A (ω )Zt (ω, ω )dPt (ω, dω ). Observe that k˜ t (ω, ·) is absolutely continuous with respect to Pt (ω, ·) and that we have for all A ∈ P j,m with j2−m ≤ t k˜ t (ω, A) = kt (ω, A) for PM −a.a. (ω, t) ∈ Ω × [0, 1]. By integrating, we obtain the equation

t

Pt (ω, A) = P(A) +

(21)

0

k˜ s (ω, A) dMs + LA t (ω)

for all A ∈ j2−m ≤t P j,m . Since the LHS and both expressions on the RHS are measures coinciding on a system which is stable for intersections, Eq. (21) holds for all A ∈ G0t− . Hence, by choosing kt (·, A) = k˜ t (·, A) for all A ∈ G0t− , the proof is complete. We close this section by illustrating the method developed by means of an example. Example 3.1. Let W be the Wiener process, P the Wiener measure, F t0 the filtration generated by W, a > 0, τ(a) = 1 ∧ inf{t ≥ 0 : W t = a}, δ > 0,

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16

Ht0 = σ(τ ∧ t + δ) and G0t = Ft0 ∨ Ht0 . Again let (Ft ) and (Gt ) be the smallest respective extensions of (Ft0 ) and (G0t ) satisfying the usual conditions. An investor having access to the information represented by (Gt ) knows at any time whether within the next δ time units the Wiener process will hit the level a, provided the level has not yet been hit. In this example, the information drift of (Gt ) is already completely determined as the density process of kt (ω, ·) relative to Pt (ω, ·) along the σ−algebras Ht0 (this follows from a slight modification of the proof of Theorem 3.1). Let S = sup0≤r≤t Wr , F(a, x, u) = P(τ(a − x) ≤ u) and recall that F(a, x, u) = u y t (a−x)2 √ exp(− 2y )dy, for all x < a (see Ch.III, p.107 in [25]). Note that 0 2πy3

for all r ≤ u ≤ 1 we have Pr (ω, {τ(a) ≤ u}) = 1{Sr ≥a} + 1{Sr u r s and we denote by µs the information drift of (Ku ) relative to M. The conditional entropy of the σ−algebra G 0s relative to the filration (Fu0 ) on the time interval [s, t], t ∈ (s, 1], will be defined by H (s, t) = HG0s (Pt (ω, ·)Ps(ω, ·))dP(ω). We will now show that 2 H (s, t) is equal to the square-integral of µs on Ω × [s, t]. To this end let (Pm )m≥0 be an increasing sequence of finite partitions such that σ(Pm : m ≥ 0) = G0s . Then H (s, t) = HG0s (Pt (ω, ·)Ps (ω, ·))dP(ω)

=E 1A log Ps (·, A) − 1A log Pt (·, A) A∈Pm

"

=E

− s

A∈Pm

+

1 2

t s

t

ku Pu

2

ku ˜u− (·, A) 1A dM Pu

t s

ku (·, A) 1Aµsu dM, Mu Pu

# (·, A) 1A dM, Mu ,

˜ is a local martingale, where the last equation follows from (20). Since M we obtain by stopping and taking limits if necessary ⎡ ⎢ ⎢⎢ H (s, t) = E ⎢⎣ A∈Pm

t s

ku 1 (·, A) 1Aµu dM, Mu − Pu 2

t s

ku Pu

2

⎤ ⎥⎥ (·, A) 1A dM, Mu ⎥⎥⎦ .

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18

2 $ Lemma 3.1 implies that A∈Pm Pkuu (ω, A) 1A (ω ) is an L2 (Pu (ω, ·))-bounded martingale for PM −a.a. (ω, u), and therefore, by Theorem 3.1 t 2 t ku (·, A) 1A dM, Mu = E (µsu )2 dM, Mu . lim E m P u s s m A∈P

Similarly we have t t ku (·, A) 1A µsu dM, Mu = E (µsu )2 dM, Mu . lim E m P u s s m A∈P

and hence 1 H (s, t) = E 2

(22)

s

t

(µsu )2 dM, Mu .

We are now in a position to introduce a notion of conditional entropy between our filtrations (G0t ) and (Ft0 ). For any partition ∆ : 0 = t0 ≤ t1 ≤ $ $ % % . . . ≤ tk = 1 we will use the abbreviations ∆ = ki=1 and ∆ = ki=1 Definition 4.1. Let (∆ n ) be a sequence of partitions$of [0, 1] with mesh |∆n | converging to 0 as n → ∞. The limit of the sums ∆n H (ti−1 , ti ) as n → ∞ is called conditional entropy of (G 0t ) relative to (Ft0 ) and will be denoted by HG0 |F 0 . Theorem 4.1. The conditional entropy H G0 |F 0 is well defined and it satisfies 1 1 µ2u dM, Mu . HG0 |F 0 = E 2 0 Proof. Let (∆n ) be a sequence of partitions of [0, 1] with mesh |∆| converging to 0 as n → ∞. For all ∆n we define auxiliary filtrations & (Fs0 ∨ G0ti ) if t ∈ [ti , ti+1 [. Dnt = s>t

Since all (Dnt ) are subfitrations of (G0t ), the respective information drifts µn of M exist. It follows immediately from Eq. (22) that t 1 H (ti−1 , ti ) = E (µnu )2 dM, Mu . 2 s n ∆

As it is shown in Theorem 4.4 in [4], the information drifts µn converge in L2 (M) to the information drift µ. Consequently, the conditional entropy of 1 (G0t ) relative to (Ft0 ) is well defined and equals 12 E 0 µ2u M, Mu .

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19

The conditional entropy HG0 |F 0 can be interpreted as a multiplicative integral along the filtration (G0t ). More precisely, if for any s ≤ t ≤ 1 we define P (ω,·) d(s, t, ω, ω ) = t (ω ), and if ∆ is a partition of [0, 1], then Ps (ω,·) G0 s

H (ti−1 , ti ) =

∆

∆

=

∆

=

log

log

Pti (ω, ·) (ω )P (ω, dω ) dP(ω) ti Pti−1 (ω, ·) G0ti−1

log d(ti−1 , ti , ω, ω)dP(ω) '

d(ti−1 , ti , ω, ω)dP(ω)

∆

In the special case where (G0t ) is obtained by an initial enlargment with a Pt (ω,·) Pt (ω,·) random variable G, we have Ps (ω,·) 0 = Ps (ω,·) and hence Gs σ(G) P1 (ω, dω ) HG0 |F 0 = log (ω )P1 (ω, dω ) dP(ω) P(dω ) σ(G) = HF1 ⊗σ(G) (P1 (ω, dω )P(dω)P ⊗ P). The image of the measure P1 (ω, dω )P(dω) under the mapping (ω, ω ) → (M(ω), G(ω )) is the joint distribution of M = (Mt )0≤t≤1 and G. Consequently, in the initial enlargement case, HG0 |F 0 is equal to the entropy of the joint distribution of M and G relative to the product of the respective distributions, which is also known as the mutual information between M and G. To sum up, we obtain a very simple formula for the additional logarithmic utility under initial enlargements. ! Theorem 4.2. Let G be a random variable and G t = s>t Fs ∨ σ(G). Then uG (x) − uF (x) coincides with the mutual information between M and G. References 1. J. Amendinger, D. Becherer, and M. Schweizer. A monetary value for initial information in portfolio optimization. Finance Stoch., 7(1):29–46, 2003. 2. J. Amendinger, P. Imkeller, and M. Schweizer. Additional logarithmic utility of an insider. Stochastic Process. Appl., 75(2):263–286, 1998. 3. S. Ankirchner. Information and Semimartingales. Ph.D. thesis, Humboldt Universität Berlin, 2005. 4. S. Ankirchner, S. Dereich, and P. Imkeller. The shannon information of filtrations and the additional logarithmic utility of insiders. Annals of Probability, 34:743–778, 2006. 5. S. Ankirchner and P. Imkeller. Finite utility on financial markets with asymmetric information and structure properties of the price dynamics. Ann. Inst. H. Poincaré Probab. Statist., 41(3):479–503, 2005.

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6. F. Baudoin. Conditioning of brownian functionals and applications to the modelling of anticipations on a financial market. PhD thesis, Université Pierre et Marie Curie, 2001. 7. F. Biagini and B. Oksendal. A general stochastic calculus approach to insider trading. Preprint, 2003. 8. L. Campi. Some results on quadratic hedging with insider trading. Stochastics and Stochastics Reorts, 77:327–248, 2003. 9. J. Corcuera, P Imkeller, A. Kohatsu-Higa, and D. Nualart. Additional utility of insiders with imperfect dynamical information. Preprint, September 2003. 10. F. Delbaen and W. Schachermayer. The existence of absolutely continuous local martingale measures. Ann. Appl. Probab., 5(4):926–945, 1995. 11. C. Dellacherie and P.-A. Meyer. Probabilities and potential, volume 29 of NorthHolland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1978. 12. D. Duffie and C. Huang. Multiperiod security markets with differential information: martingales and resolution times. J. Math. Econom., 15(3):283–303, 1986. 13. H. Follmer ¨ and P. Imkeller. Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. H. Poincaré Probab. Statist., 29(4):569–586, 1993. 14. D. Gasbarra and E. Valkeila. Initial enlargement: a bayesian approach. Theory of Stochastic Processes, 9:26–37, 2004. 15. A. Grorud and M. Pontier. Insider trading in a continuous time market model. International Journal of Theoretical and Applied Finance, 1:331–347, 1998. 16. P. Imkeller. Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus. Probab. Theory Related Fields, 106(1):105–135, 1996. 17. P. Imkeller. Random times at which insiders can have free lunches. Stochastics and Stochastics Reports, 74:465–487, 2002. 18. P. Imkeller. Malliavin’s calculus in insider models: additional utility and free lunches. Math. Finance, 13(1):153–169, 2003. Conference on Applications of Malliavin Calculus in Finance (Rocquencourt, 2001). 19. P. Imkeller, M. Pontier, and F. Weisz. Free lunch and arbitrage possibilities in a financial market model with an insider. Stochastic Process. Appl., 92(1):103–130, 2001. 20. J. Jacod. Grossissement initial, hypothese (H’), et théorème de Girsanov. In Th. Jeulin and M. Yor, editors, Grossissements de filtrations: exemples et applications, pages 15–35. Springer-Verlag, 1985. 21. Th. Jeulin. Semi-martingales et grossissement d’une filtration, volume 833 of Lecture Notes in Mathematics. Springer, Berlin, 1980. 22. Th. Jeulin and M. Yor, editors. Grossissements de filtrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985. Papers from the seminar on stochastic calculus held at the Université de Paris VI, Paris, 1982/1983. 23. K.R. Parthasarathy. Introduction to probability and measure. Delhi etc.: MacMillan Co. of India Ltd. XII, 1977. 24. I. Pikovsky and I. Karatzas. Anticipative portfolio optimization. Adv. in Appl.

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Probab., 28(4):1095–1122, 1996. 25. D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. 26. M. Yor. Entropie d’une partition, et grossissement initial d’une filtration. In Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.), volume 1118 of Lecture Notes in Math. Springer, Berlin, 1985. 27. M. Yor. Grossissement de filtrations et absolue continuité de noyaux. In Grossissements de filtrations: exemples et applications. T. Jeulin, M.Yor (eds.), volume 1118 of Lecture Notes in Math. Springer, Berlin, 1985. 28. M. Yor. Some aspects of Brownian motion. Part II. Lectures in Mathematics ETH Zurich. ¨ Birkhäuser Verlag, Basel, 1997. Some recent martingale problems.


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ritsuhregloc

A Localization of the Lévy Operators Arising in Mathematical Finances Mariko Arisawa GSIS, Tohoku University, Aramaki 09, Aoba-ku, Sendai 980-8579, Japan

1. Introduction We study the uniform Holder ¨ continuity of the solutions of the following problem. F(x, ∇v(x), ∇2v(x)) − [v(x + z) − v(x) RN

(1)

−1|z|0

which according to (32) corresponds to the moment generating function of a 2a Gamma distribution for X0 with parameters σ2 , 1 . Then (39)

2a

χTn (φ) = cn (φ Hn + Kn )− σ2 −n pn (φ)

where Hn and Kn satisfy the recursions ⎧ ⎪ Hn = Rn Hn−1 + Un Kn−1 , H0 = 1 ⎪ ⎪ ⎨ (40) ⎪ ⎪ ⎪ ⎩ Kn = Sn Hn−1 + Vn Kn−1 , K0 = 1 the coefficient cn is given by (41)

# 2a $−1 − 2 −n cn = Kn σ pn (0)

and pn (φ) is a polynomial of degree n − 1 given by (42) ⎧ 1 for n = 0 and n = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2a ⎪ ⎪ − pn (φ) 2 − n + 1 Hn (φUn + Vn ) % ⎪ σ ⎪ ⎨ pn (φ) = ⎪ ⎪ ⎪ ⎪ ∂ % ⎪ pn (φ) pn (φ) + (φHn + Kn )(φUn + Vn ) ∂φ +Un (φHn + Kn )% ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ for n ≥ 2 with (43)

! % pn (φ) = (φUn + Vn )

n−2

pn−1

" φRn + Sn , n≥2 φUn + Vn

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10:57


rung

112

Proof : The statement is clearly true for n = 0. We show it first for n = 1 and then inductively for all n ≥ 2. i) the case n = 1 : by (35), (36), (37) and the recursions in (40) we have ⎧ ⎫ 2a ⎪ ⎪ ⎪ ⎪ βR +S ⎨ W1 ⎬ σ2 χT0 βU1 +V1 ⎪ ⎪ +V βU ⎪ 1 1 ⎪ ⎩ 1 1 ⎭ |β=φ ⎧ ⎫ 2a ⎪ ⎪ ⎪ W1 βR1 +S1 ⎪ ⎬ σ2 ∂ ⎨ χT0 βU +V ⎪ ⎪ ∂β ⎪ 1 1 ⎪ ⎩ βU1 +V1 ⎭

∂ ∂β

χT1 (φ) =

|β=0

(44)

⎧ ⎫ 2a − 2a ⎪ ⎪ ⎪ ⎪ ⎨ W1 σ2 βH1 +K1 σ2 ⎬ ⎪ ⎪ ⎪ ⎪ βU1 +V1 ⎩ βU1 +V1 ⎭ |β=φ ⎧ ⎫ 2a 2a ⎪ ⎪ − ⎪ ⎪ 2 2 +K W βH ⎨ ⎬ σ σ ∂ 1 1 1 ⎪ βU1 +V1 ⎪ ⎪ ∂β ⎪ ⎩ βU1 +V1 ⎭

∂ ∂β

=

=

− 2a −1 σ2

(φH1 +K1 )

− 2a −1 σ2

K1

|β=0

which indeed corresponds to (39) with (41) and (42). ii) the general case n ≥ 2 : assume (39) holds for n − 1. Then, always by (35), (36), (37) and (40) we obtain χTn (φ) =

∂ ∂β ∂ ∂β

(45) =

∂ ∂β

∂ ∂β

Wn βUn +Vn Wn βUn +Vn

1 βUn +Vn 1 βUn +Vn

2a

σ2

2a

σ2

χTn−1 χTn−1

βR

βR

n +Sn βUn +Vn n +Sn βUn +Vn

|β=φ

|β=0

2a βH

σ2

σ2

2a βR +S n +Kn − σ2 −n+1 pn−1 βUnn +Vnn βUn +Vn

2a βH

2a βR +S n +Kn − σ2 −n+1 pn−1 βUnn +Vnn βUn +Vn

|β=φ

|β=0

Taking into account that, by (43), ! " βRn + Sn (46) pn−1 = (βUn + Vn )2−n% pn (β) βUn + Vn the numerator in the rightmost expression of (45) becomes (47) − 2a2 −n+1 ∂ σ % pn (β) ∂β (βUn + Vn )(βHn + Kn ) |β=φ

2a = (βHn + Kn )− σ2 −n Un (βHn + Kn )% pn (β) ∂ % + − σ2a2 − n + 1 Hn (βUn + Vn )% pn (β) pn (β) + (βUn + Vn )(βHn + Kn ) ∂β

|β=φ

2a

= (φ Hn + Kn )− σ2 −n pn (φ)

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rung

113

where we have used the definition of pn (φ) in (42). Returning to (45) and recalling that the denominator in the rightmost expression of (45) is the same as the numerator except for putting β = 0, one finally obtains 2a

(48)

χTn (φ) =

(φ Hn + Kn )− σ2 −n pn (φ) − 2a2 −n Kn σ pn (0)

2a

= cn (φ Hn + Kn )− σ2 −n pn (φ)

Remark 14. It follows from Theorem 13 that, for a choice of the initial distribution corresponding to χ0 (φ) in (38), the sequence χTn (φ) and therefore (see Steps iii) and iv) in Section 4.3) the entire filter is parameterized by a same finite number of sufficient statistics, namely the pairs (Hn , Kn ) and the polynomial functions pn (φ), all of which can be computed recursively on the basis of the functions Rn , Sn , Un , Vn of the interarrival times of the defaults. References 1. Bielecki, T. R., M. Jeanblanc, & M. Rutkowski (2004). Modeling and Valuation of Credit Risk. In : Stochastic Methods in Finance (M. Frittelli and W. Runggaldier, eds), Lecture Notes in Mathematics. Vol 1856, 27–126. Springer Verlag. 2. Ceci, C., & A. Gerardi (2006). A Model for High Frequency Data under Partial Information : a Filtering Approach. International Journal of Theoretical and Applied Finance, 1–22. 3. Duffie, D., & N. Gârleanu (2001). Risk and Valuation of Collateralized Debt Obligations. Financial Analysts Journal. 57, 41–59. 4. Frey, R., & W. J. Runggaldier (2006). Credit Risk and Incomplete Information : a Nonlinear Filtering Approach. Preprint. 5. Kliemann, W. H., G. Koch, & F. Marchetti (1990). On the Unnormalized Solution of the Filtering Problem with Counting Process Observations. IEEE Transactions on Information Theory. 36, 1415–1425. 6. Lamberton, D., & B. Lapeyre (1995). An Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall. 7. McNeil, A. J., R. Frey, & P. Embrechts (2005). Quantitative Risk Management. Princeton University Press. 8. Schonbucher, ¨ P. (2004) Information-driven Default Contagion. Preprint, Department of Mathematics, ETH Zurich. ¨


December 26, 2006

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Smooth Rough Paths and the Applications Keisuke Hara1∗and Terry Lyons2 1

Department of Mathematical Sciences, Ritsumeikan University, Japan 2 Mathematical Institute, University of Oxford, UK

Key words: p-variation, rough path

1. Introduction This article is based on a joint work submitted to a journal (K. Hara and T. Lyons [1]), which was mainly studied in the first author’s academic year 2004–2005 in Oxford. In this article, we will show the idea, the main results, and the sketch of the proofs. We will also show the related problems and some examples, which are not included in our paper mentioned above.

First of all, we explain the essence of our idea: smooth rough paths. It is a good starting point to recall the well known definition of p-variations. Let p ≥ 1 be a real number and F : I → R n be a continuous function on an interval I in R. Then we define the p-variation (norm) of F by Fp−var,I

⎧ ⎫1/p ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎨ p⎬ sup =⎪ |F(t ) − F(t )| , ⎪ j+1 j ⎪ ⎪ ⎪ ⎪ ⎩ D j=1 ⎭

where the sup runs over all finite dissection {t j } of the interval I. If the interval I is finite, this definition should interest us only for non-smooth paths like Brownian paths because smooth paths have the trivial estimate with the derivatives like |F(t j+1 ) − F(t j )| ≤ C|F (t j )|(t j+1 − t j ). However, what happens if I is the whole real line R ? Now the difference |F(t j+1 ) − F(t j)| can easily sum up to the infinite even if the path is smooth. Therefore, we can ask when we have the finite p-variation for smooth paths. In other words, we can study how smooth paths oscillate globally with their p-variations. ∗

Partially supported by ACCESS Co. Ltd. 115

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The concept of smooth rough paths is a generalization of this idea. More precisely, we ask not only the p-variations of paths themselves but also the variations of the iterated integrals of the paths in the framework of rough path theory. By this procedure, we can get more information how they oscillate globally and specially how they behave at the infinities. For example, we can apply the general framework of rough path theory to study the differential equations driven by the path if we establish the rough path property of the smooth path. In the following section, we will give simple definitions that we need. The main results and the sketch of the proofs will be shown in Section 3. Section 4 and 5 are devoted to the applications to Fourier analysis and the related problems. 2. Definitions We prepare the basic concept of rough path theory. Though rough path theory has a very general framework, the almost definitions here can become much simpler than general rough path theory because we assume that the paths are smooth. Let F(t) : I → Rn be a continuous function defined on a closed interval I = [a, b] (where a or b may be infinite, +∞ or −∞). We are interested in the oscillation of the iterated integrals: i dFt1 ⊗ · · · ⊗ dFti , i = 1, 2, . . . Fu,v = u In . Hence, (iii) is proved. Similarly, under the condition that 2 vr > In , QAB ” (Y) < 0 and A∗ β1 2 vr − In > 0. Hence, (iv) and (v) are proved. QAB YA∗ = 1 − YYB∗ (2) /Π (2) (Ip + In )−Ip, To sum, under the conditions that (In /2) < vr ≤ Π the access-to-bypass equilibrium exists and YL∗ is unique. Proof of Proposition 5.1 (i) Two types of equilibrium can emerge in service-based competition, depending on the level of access charge and the degree of positive network externality; the one is the access-to-bypass equilibrium and the other is bypass equilibrium. The bypass equilibrium emerges when a follower adopts the bypass strategy. In fact, the trigger point of a follower in the bypass equilibrium is exactly the same as in the facility-based competition equilibrium, YB . Hence we need to check the trigger points of a follower only in the access-to-bypass equilibrium. In the access-to-bypass equilibrium, the relationship that YA∗ < YB∗ holds. Then, as shown in the proof of Lemma 1, we can confirm that YA∗ < YB∗ implies YA∗ < YB . (ii) We show that YB < YB∗ if and only if YA∗ < YB∗ . This is because β1 r − α n v β1 r − α p v < ≡ YB∗ I + YA∗ ≡ I − β1 − 1 Π r β1 − 1 ∆Π (2) r (2) (2) (Ip + In ) < Π (2) In − v ⇔ Π (2) (Ip + In ) − Π r β β r − α r − α v 1 1 (Ip + In ) < ⇔ YB ≡ ≡ YB∗ . In − β1 − 1 Π (2) β1 − 1 ∆Π (2) r

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Therefore, we have YB < YB∗ . Proof of Proposition 5.3 Substituting (17) and (18) for ∀Y ∈ 0, YA∗ into QAB (Y) ≡ VLAB (Y) − VFAB (Y) and rearranging it, we have YΠ (1) − (Ip + In ) r−α Y β1 p v YA∗ Π (1) I +2 − + A∗ r r−α Y β1 Y v + B∗ In − 2 . r Y

QAB (Y) =

(31)

To prove the proposition, we check the sign of QAB (YL ) where YL is the trigger point of the leader under facility-based competition equilibrium. Note that YL is characterized by VLB (YL ) = VFB (YL ), so that we have β1 B β1 −1 YL Π (1) Y Π (2) YL YL − (Ip + In ) + B 1− B r−α r−α Y Y β1 B Y Π (2) YL − (Ip + In ) = B r−α Y or (32)

β1 B YL YL Π (1) Y Π (1) − (Ip + In ) = B − (Ip + In ) . r−α r−α Y

Substituting (32) into QAB (YL ) gives QAB (YL ) = (YL )β1 χ (x) , where

(33)

−β1 YB Π (1) B χ (x) ≡ Y − (Ip + In ) r−α −β1 −β1 v YA∗ Π (1) v In − 2 Ip + 2 − + YA∗ + YB∗ r r−α r −β1 β1 Π (1) p (I + In ) − (Ip + In ) = YB β1 − 1 Π (2) ⎤ ⎡ ⎢ β1 Π (1) p v ⎥⎥ v A∗ −β1 ⎢ p I + ⎥⎥⎦ + Y ⎢⎢⎣I + 2 − r β1 − 1 Π r (2) −β1 v + YB∗ In − 2 r

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(2) , Ie, Im . and x ≡ v, Π (1) , Π (2) , Π Hence QAB (YL ) < 0 if χ (x) < 0. Note that the terms in the bracket of the third term of (33) is negative. Hence, χ (x) < 0 if −β1 YB

(34) where j ≡

β1 Π (1) p (I + In ) − (Ip + In ) β1 − 1 Π (2) ⎡ ⎤ −β1 ⎢⎢ β1 Π (1) p v ⎥⎥ v A∗ p ⎢ I + ⎥⎥⎦ + Y ⎣⎢I + 2 r − β − 1 r 1 Π (2) −β1 ) −β1 * = YB [l − mΠ (1)] < 0 jΠ (1) − k + YA∗

β1 Ip +In β1 −1 Π(2) ,

k ≡ (Ip + In ), l ≡ I p + 2 vr and m ≡

β1 Ip +(v/r) . β1 −1 Π(2)

Let us

show (34) holds if Π (1) is sufficiently ) large. * Define Γ [Π (1)] ≡ [mΠ (1) − l] / jΠ (1) − k . Note that Lemma 4.1 ensures j > m as long as the access-to-bypass equilibrium exists, and l > k under the conditions of Proposition 4.1. Hence, (35)

jl − km dΓ [Π (1)] = ) *2 > 0. dΠ (1) jΠ (1) − k

Equation (35) shows that Γ [Π (1)] is an increasing function of Π (1) and that Γ [Π (1)] monotonically converges to m/ j as Π (1) goes to infinity. Let β us compare YA∗ /YB with Γ [Π (1)]. Since YA∗ /YB = m/ j < 1 and β > 1, ! "β there exists a threshold of Π (1) above which m/ j < Γ [Π (1)] or + (36)

YA∗ YB

,β
0, we have (34). Therefore, if Π (1) is sufficiently large, a leader enters earlier in the facility-based competition equilibrium than in the access-to-bypass equilibrium. Proof of Proposition 5.4 Note that the term in the brackets of the first term of (33) is positive, while that in the brackets of the third term is negative. Suppose (37)

β1 Π (1) p v v Ip + 2 − > 0. I + r β1 − 1 Π r (2)

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Then, because YA∗ < YB < YB∗ , we obtain −β1 β1 Π (1) p B∗ (I + In ) − (Ip + In ) χ (x) > Y β1 − 1 Π (2) ⎤ β1 Π (1) p v n v v ⎥⎥⎥ p +I + 2 − + I − 2 ⎥⎦ I + r β1 − 1 Π r r (2) ⎡ ⎤ p n p −β1 β1 ⎢⎢ I + I I + (v/r) ⎥⎥⎥ B∗ ⎢ = Y Π (1) ⎢⎣ (38) − ⎥. β1 − 1 Π (2) (2) ⎦ Π The right-hand side of (38) is positive because of (15). This means that (37) is a sufficient condition for YL∗ < YL . Proof of Corollary 6.1 As in the proof of Proposition 5.3, we show that χ (x) < 0 if σ is sufficiently large. (2) (Ip + In ) . On Remember that YA∗ /YB = m/ j = [Π (2) (Ip + (v/r))] / Π ) * other hand, Γ [Π (1)] [mΠ (1) − l] / jΠ (1) − k = ≡ the (2) − (Ip + 2v/r) / [BΠ (1) (Ip + In ) /Π (2) − (Ip + In )] BΠ (1) (Ip + (v/r)) /Π ! " where B ≡ β/ β − 1 . Then, as σ → +∞, β → 1 and B → +∞. Hence, β (2) (Ip + In ) = lim Γ [Π (1)]. This lim YA∗ /YB = [Π (2) (Ip + (v/r))] / Π σ→+∞

σ→+∞

means that the first term and the second term of χ (x) are cancelled out when σ is sufficiently large. Since the third term of χ (x) is always negative and approaches to zero, we can state that χ (x) < 0 if σ is sufficiently large. References 1. Biglaiser, G. and M. Riordan (2000), “Dynamic Price Regulation”, Rand Journal of Economics, 31, 744–767. 2. Bourreau, M. and P. Dogan ˘ (2005), “Unbundling the Local Loop”, European Economic Review, 49, 173–199. 3. Dixit, A. K. and R. S. Pindyck, (1994), Investment under Uncertainty, Princeton, NJ: Princeton University Press. 4. Fudenberg, G. and J. Tirole (1985), “Preemption and Rent Equalisation in the Adoption of New Technology”, Review of Economic Studies, 52, 383–401. 5. Gans, J. S. (2001), “Regulating Private Infrastructure Investment: Optimal Pricing for Access to Essential Facilities”, Journal of Regulatory Economics, 20, 167–189. 6. Gans, J. S. and P. L. Williams (1999). “Access Regulation and the Timing of Infrastructure Investment”, Economic Record, 75, 127–137. 7. Hori, K. and K. Mizuno (2006a), “Access Pricing and Investment with Stochastically Growing Demand”, International Journal of Industrial Organization, 24, 795–808.

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8. Hori, K. and K. Mizuno (2006b), “Competition schemes and investment in network infrastructure under uncertainty”, mimeo. 9. Katz, M. K. and C. Shapiro (1987), “R&D Rivalry with Licensing or Limitation”, American Economic Review, 77, 402–429. 10. Mason, R. and H. Weeds (2003), “Can Greater Uncertainty Hasten Investment?”, mimeo. 11. Nielsen, M. J. (2002), “Competition and Irreversible Investments”, International Journal of Industrial Organization, 20, 731–743. 12. Sidak, G. and D. Spulber (1997). Deregulatory Takings and the Regulatory Contract, Cambridge: Cambridge University Press. 13. Valletti, T. M. (2003), “The Theory of Access Pricing and its Linkage with Investment Incentives”, Telecommunications Policy, 27, 659–675. 14. Weeds, H. (2002). “Strategic Delay in a Real Options Model of R&D Competition”, Review of Economic Studies, 69, 729–747.

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The Investment Game under Uncertainty: An Analysis of Equilibrium Values in the Presence of First or Second Mover Advantage. ∗ Junichi Imai1 and Takahiro Watanabe2 1

Graduate School of Economics and Management, Tohoku University 2 Faculty of Urban Liberal Arts, Tokyo Metropolitan University

In this paper we develop a valuation model when there exist two competitive firms that face irreversible investment decisions under the demand uncertainty. We propose a numerical procedure to derive both project values and equilibrium strategies in the duopolistic environment. In numerical examples we consider two different economic conditions, which are labelled first mover advantage and second mover advantage, and examine the effects of these conditions on the equilibrium strategies as well as the project values. We show that these conditions cause significant changes in the equilibrium strategies of both firms. JEL: G31,C61,C73 Key words: Real option, Investment game, Equilibrium strategy, Project Valuation

1.

Introduction The real option approach valuing real assets or projects has been playing an important role in the field of corporate finance and financial economics. It has been proposed as a useful concept to analyze a strategic investment. The usual real option analysis is implicitly based on the assumption that the underlying risk is exogenous and that management cannot affect the underlying stochastic process. This assumption is appropriate if a firm is nearly in a perfectly competitive market or it has monopolistic power over the market of the project. However, management in the real world ∗ Both authors acknowledge a research support from Grant-in-Aid for Scientific Research in Japan Society for the Promotion of Science 17510128.

151

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should consider rival firms’ behaviors in an imperfectly competitive market because one firm’s action affects other firms’ decisions and vice versa. Therefore, we should consider strategic interaction among competitive firms. Several studies recently focus on integrating real option analysis with game theory to reflect the interaction. In this paper we develop a valuation model that can derive the project values and the equilibrium strategies when there exist two competitive firms that face irreversible investment decisions under uncertainty. Suppose each firm operates an existing project that generates a cash flow stream. Each firm has an option to reinvest in the project so that the marginal cash flow of the project is increased. The cash flow stream from the project depends on the level of demand that follows a continuous-time stochastic process, and the investment decisions of both firms. Each firm makes an investment decision to maximize its project value. We approximate the underlying continuous-time process with the discrete-time lattice process and propose a numerical procedure to derive the project values in equilibrium strategies. In numerical examples we derive the project values and analyze the strategic behavior of the competitive firms in the equilibrium under different economic situations. Recent researches that integrate real option approach with game theory can be classified into two categories. The models in the first category assume that the underlying variable of the project follows a one-period or two-period process. Smit and Ankum (1993), Kulatilaka and Perotti (1998), Smit and Trigeorgis (2001) are in the first category. These studies give us intuitive ideas about the effect of the competition under uncertainty and clarify the strategic behaviors of the firms under competition. However, several strong assumptions are contained in the models. Especially, the underlying process is too simple and the number of decision opportunities is strictly restricted. As a result, they are not suitable for a quantitative analysis. On the other hand, the models in the second category often assume that the underlying variable follows a continuous-time stochastic process. Smets (1991) and Dixit and Pindyck (1994) are examples of the earliest studies in this category. The models are constructed by extending a deterministic model proposed by Fudenberg and Tirole (1985) to a stochastic dynamic process. Grenadier (1996) applies this model to an analysis of land development projects while Huisman (2001) refines it to analyze the strategic behaviors of the firms. Other studies in this category are Hoppe (2000), Lambrecht (2000), Grenadier (2002), Huisman and Kort (2003) and Huisman and Kort (2004). Although the models in the second category can derive valuable implication about strategic investments under uncertainty and competition,

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several assumptions in these models seem to be too naive to apply to a practical use of evaluating real projects. First, the assumption that the project has an infinite horizon is not always realistic and the models does not capture the effect of the maturity on the project value. Secondly, most researches assume that the underlying variable follows a geometric Brownian motion1 . If we assume the different underlying process or loosen the infinite horizon assumption it is not easy to derive a closed-form solution of the problem. Accordingly, a numerical approach is valuable in this case. The third assumption is about the continuity of the decision opportunities. Most studies implicitly assume that management can make investment decisions continuously. It is observed, however, that these decision opportunities do not always come continuously even if the underlying risk would change continuously. They come once in a week or in a month, sometimes in a quarter in many actual investment projects. Therefore, it is necessary to construct a model to distinguishes the continuity of the underlying process from the discrete decision opportunities. The purposes of this paper is (1) to derive a numerical procedure to evaluate an investment project under the demand uncertainty and competition; (2) to analyze the project values and the equilibrium strategies under two conditions of competitive environments, which is called First Mover Advantage (FMA) and Second Mover Advantage (SMA). For the first purpose we develop a discrete-time lattice model and integrate real option approach with game theory. It is well known that the lattice process can approximate the continuous-time stochastic process if the parameter values are adequately adjusted. Furthermore, this model can be easily extended to deal with the finite number of decision opportunities under a wider class of continuous-time underlying processes. Our model is located between the two categories stated above; namely, our model can assume a continuous-time underlying process with finite number of decision opportunities, which enables us to derive more realistic results and implication. Although our numerical approach cannot obtain analytical solutions, we can obtain the current project value that depends on both the current demand and time to the project maturity, which have not been analyzed by the past studies. Imai and Watanabe (2006b) propose a similar numerical procedure. In their model, however, two firms are assumed to be asymmetric where the two firms make an investment decision sequentially: one firm can make a decision first at each period and the other firm does after observing the rival’s decision. This corresponds to a problem of a perfect information in game theory. The game is simple because a multiple equilibria problem 1 Kijima

and Shibata (2002) consider the stochastic volatility process.

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never occurs and it can be solved by a simple backward induction. The research with regard to symmetric and simultaneous decisions is also necessary because it is a more fundamental setting from a theoretical viewpoint. It also enables us to compare the results of the existing researches such as Smets (1991), Dixit and Pindyck (1994), Grenadier (1996) and Huisman (2001). Therefore, in our model, both firms make investment decisions simultaneously at each decision opportunity. Since multiple equilibria could emerge in the simultaneous decision case we introduce the following two criteria to select an equilibrium. In the first criterion, we distinguish the two competitive firms before starting the game, which are labelled firm L and firm F in the paper. Then we assume that firm L has a competitive advantage over firm F in the sense that one equilibrium is selected so that the project payoff of the firm L is maximized if there exist multiple equilibria. We call it firm L advantage criterion (LAC) for convenience in the paper. Although LAC looks rather ad hoc Imai and Watanabe (2006a) point out that LAC is consistent with a theory of equilibrium selection in game theory developed by Harsanyi and Selten (1988) if we suppose that the cash flow of firm L is infinitesimally greater than that of firm F after both firms’ investment. In the second criterion, we assume that one equilibrium is selected with an equal probability when there exist multiple equilibria. This criterion is consistent with the previous studies of Grenadier (1996) and Huisman (2001) that examine a preemption game under the assumption of FMA. The criterion is exogenously assumed in our paper, but Fudenberg and Tirole (1985) derive the same criterion endogenously with a more sophisticated model. In this paper we call it 50% criterion. The second purpose of this paper is motivated as follows. Recent studies concerning the integration of real option approach with game theory mainly focus on investments of two competitive firms under the condition of FMA. FMA is a condition in which the increment of the marginal cash flow of a firm that invests earlier is greater than that of the other firm that invests later. In this paper a firm which invests in the project earlier is labelled the first mover while a firm which invests after the first mover is labelled the second mover. It is known that the condition of FMA causes a preemption game because each firm wants to invest earlier than the rival firm in order to prevent the rival firm’s investment and to enjoy the monopolistic revenue from the investment. In other words, both firms want to become the first mover of the game. FMA is an appropriate condition to describe preempt competition under uncertainty such as land developments and oil refinery projects. However, some competitive situations cannot be captured by the context of FMA. For example, consider an investment opportunity to enter a new

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product market. The leader firm, which intends to launch a new product, often needs to invest in the research and development. The marketing cost is also necessary for the promotion of the new product to create a new market. On the other hand, the follower firm, which intends to enter the market after the leader, could learn from the leader’s experience. The follower can observe the market development and decide to invest after the market is matured. Furthermore, the follower has to pay less money for the market creation. To describe this situation SMA is a more appropriate condition where the increment of cash flow of the second mover is larger than that of the first mover. In this paper, we examine the equilibrium strategy of the competitive firms under the condition of SMA as well as that of FMA. In numerical examples we analyze comparative statics under the conditions of both FMA and SMA with respect to the initial demand and examine the effect of these two conditions on the equilibrium strategies as well as the project values. We observe that a preemption game emerges under the condition of FMA that leads to multiple equilibria. As a result, an asymmetric equilibrium where only firm L invests first, is chosen in case of LAC. In that case the project value of firm L, which is the first mover, is larger than that of firm F. However, we can conclude that the project value of firm L is not always larger than the project values accomplished by both firms in coordination. This is because the firms tends to invest earlier than the optimal timing of the firm that is assigned to be the first mover exogenously. The equilibrium strategies are always symmetric under the condition of SMA. Both firms can make the best use of the flexibility to defer the investment even in the presence of competition. As a result the project values of both firms are equivalent to the coordinated project value. This paper is organized as follows. In section 2 we develop a valuation model. Numerical analyses are done in section 3. Finally, concluding remarks are in section 4. 2. Model Description 2.1 A Basic Mode This section provides a valuation model of the project under demand uncertainty and competition. The model is based on Dixit and Pindyck (1994), Grenadier (1996) and Huisman (2001) which can be regarded as one of the typical models for the project valuation under uncertainty and competition. Two firms are introduced, denoted by firm L and firm F. Each firm operates a project, which has an option to reinvest in the project that increases the firm’s cash flow by paying the investment cost I. In the model we assume that the future demand is uncertain and follows a continuous-time stochastic process. Let Y(t) denote the realized demand

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at time t and it follows (1)

dY(t) = µ (t, Y(t)) dt + σ (t, Y(t)) dzP ,

where µ (t, Y(t)) and σ (t, Y(t)) are the instantaneous drift and volatility, respectively, and dzP is the increment of the Wiener process under the probability measure P. As explained in the introduction, our model is established on the lattice process that can converge to the corresponding continuous-time process as the number of periods tends to infinity. In addition, the lattice model can deal with finite decision opportunities under the continuous-time stochastic process and it enable us to distinguish decision opportunities from the timings of changing demand. Imai and Watanabe (2006b) analyze the effect of the discrete decision opportunities thoroughly. In this paper we do not focus on this aspect and simply assume that the decision opportunities come continuously. We assume that the projects of both firms continue within finite horizon and end up at some future time T which are called a maturity of the project. Thus, each firm can choose a time of the investment under uncertainty of the demand until the maturity of the project. To construct a lattice model, T we divided the time interval [0, T] into M periods of length ∆t: ∆t = M . The demand Y(t) changes at time t = m∆t for m = 0, · · · , M and we simply denote the demand Y(m∆t) at period m by Y(m). Note that M is assumed to be sufficiently large for the approximation. The chance of the investment in the project for each firm is at most once in the M + 1 decision opportunities and the firm can never invest again after the investment. Thus, there are two states for each firm with regard to the firm’s decision. Let xi (m) denote the state of firm i(i = L, F) at each period m = 0, · · · , M. xi (m) = 0 represents the state where firm i has not invested in the project until period m, while xi (m) = 1 represents the state where firm i has already invested. Note that xi (m) stands for the state of firm i after its decision of the investment at period m. There are totally four possible pairs of states, which are denoted by (xL (m), xF (m)). At the m-th decision opportunity for each m = 0, · · · , M when both firms have already invested (i.e., (xL (m − 1), xF (m − 1)) = (1, 1)), both firms do not make any decision. When one firm has already invested but the other firm has not yet (i.e., (xL (m − 1), xF (m − 1)) = (0, 1) or (1, 0)), only the firm that has not invested makes a decision. Finally, when neither firm has invested (i.e., (xL (m − 1), xF (m − 1)) = (0, 0)), both firms make their investment decisions. In this case, we assume that both firms determine their strategies simultaneously. The marginal cash flow obtained by each firm at each period depends on the following two variables. The first one is Y(m), the realized demand at period m. The second variable is a pair of the states of both firms about

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Firm F Invest

Fi rm L

Invest

Not invest

Not invest

( D11 , D11 )

( D10 , D01 )

( D01 , D10 )

( D00 , D00 )

Figure 1. Marginal cash flow of the project. When both firms have already invested in the project the cash flow obtained by each firm at each period is given by D11 Y(m)∆t. It is given by D00 Y(m)∆t when neither firm has invested yet. The cash flow obtained by firm L is given by D10 Y(m)∆t when only firm L finishes the investment. The cash flow obtained by firm F is given by D01 Y(m)∆t in this case. On the other hand, when only firm F has invested in the project the cash flow obtained by firm L and firm F are given by D01 Y(m)∆t and D10 Y(m)∆t, respectively.

the investment (xL (m), xF (m)). Let Dxi (m)x j (m) denote the cash flow per unit of demand of firm i where j is the rival firm of firm i. The cash flow per unit of demand and unit of period is illustrated in Fig. 1. In the usual real option analysis no-arbitrage principle is often assumed for the valuation. It is difficult, however, to apply this principle to our model since the demand of the merchandise cannot be observed in the market. Alternatively, Cox and Ross (1976), Constantinides (1978), and McDonald and Siegel (1984) propose the equilibrium approach for the option pricing. In this paper, we apply the equilibrium approach and assume that the demand risk can be considered private risk or unsystematic risk that is independent of the market risk. Since an investor pays no risk premium with respect to the unsystematic risk in equilibrium, we can assume that firms are risk neutral in the valuation model and let r denote the risk-free rate. Now, we develop the valuation model from a game theoretical perspective. First, we define a strategy of each firm. A decision whether each firm has invested or not until each period, denoted by 1 or 0, respectively, is called an action of the firm. We apply the Markov subgame perfect equilibrium to a solution concept. This means that each firm’s action depends only on the current demand and the pair of states instead of the history of their actions and the demands. The firm i’s strategy si for i = L, F is defined as a list of the actions at each period for any current demand and any state of both firms. si (m, x j (m − 1), Y(m)) is the firm i’s strategy at period m when the realized demand is Y(m) and the previous state of the rival firm j is x j (m − 1). si (m, x j (m − 1), Y(m)) = 1 indicates that firm i finishes investing until period m while si (m, x j (m − 1), Y(m)) = 0 represents that firm i has not invested yet. Now we define the present value of the project on the condition that

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the future demand is given at period m. We introduce additional notation to define it. Let Y m = (Y(m + 1), · · · , Y(M)) be the sequence of random variables for future demands after period m. Suppose that sL , sF and Y m is determined for a given period m. Then, the state of each firm i for any period l ≥ m is specified, denoted by xî (l, si , s j , xi (m − 1), x j (m − 1), Y m ). In addition, whether firm i invests in the project exactly at period l for l ≥ m is given by indicator functions 1i , which is formally written as: 1i (l,si , s j , xi (m − 1), x j (m − 1), Y m )   1 xî (l − 1, si , s j , xi (m − 1), x j (m − 1), Y m ) = 0   =  and xî (l, si , s j , xi (m − 1), x j (m − 1), Y m ) = 1,   0 otherwise. Now, let ui (m, si , s j , xi (m − 1), x j (m − 1), Y m ) be the present value of the project of firm i. We set that the cash flow obtained by each firm i at maturity M is always equal to zero, i.e., ui (M, si , s j , xi (M − 1), x j (M − 1), Y M ) ≡ 0. The project value at period m (0 ≤ m ≤ M − 1) is defined by ui (m, si , s j , xi (m − 1), x j (m − 1), Y m ) M−1 X e−r∆t(l−m) {Dxî xˆ j Y(l)∆t − I1i (l, si , s j , xi (m − 1), x j (m − 1), Y m )}, = l=m

where xî and xˆ j are precisely written as xî ≡ xî (l, si , s j , xi (m − 1), x j (m − 1), Y m ), xˆ j ≡ xˆ j (l, s j , si , x j (m − 1), xi (m − 1), Y m ). Next, we define the expected value of the project at m-th decision opportunity which is the expected cash flow after period m for a given strategy profile. The firm i’s expected value at m-th decision opportunity for each m = 0, · · · , M, denoted by Ui (m, si , s j , xi (m − 1), x j (m − 1), y), is given by Ui (m, si , s j , xi (m − 1), x j (m − 1), y) = EQ [ui (m, si , s j , xi (m − 1), x j (m − 1), Y m )|Y(m) = y]. where EQ [·] represents an expectation under the risk neutral probability measure Q. Finally, we define a concept of an equilibrium. A couple of strategy (s∗L , s∗F ) is said to be an equilibrium if and only if for any firm i = L, F, any m = 0, · · · M, any current demand y, and any states of both firms at previous period m − 1, xi (m − 1) and x j (m − 1), s∗i satisfies (2) Ui (m, s∗i , s∗j , xi (m − 1), x j (m − 1), y) ≥ Ui (m, si , s∗j , xi (m − 1), x j (m − 1), y) for any strategy si .

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2.2

Derivation of the Project Values One of our main interests is to derive the project values of the both firms in equilibrium. We define an equilibrium in the previous subsection by Eq. (2). This definition, however, is not written as a tractable form so that we cannot derive the equilibrium strategies with the definition. In this subsection we develop a dynamic programming procedure to obtain the project values of the both firms in equilibrium strategies. The recursive procedure is constructed by induction. (x (m−1),xF (m−1)) (x (m−1),xF (m−1)) Let VL L (m, y) and VF L (m, y) denote the project values of firm L and firm F, respectively. The value functions depend on period m , the current demand Y(m) = y, and the states of both firms xL (m − 1) and xF (m − 1) at period m − 1, assuming that the both firms follow equilibrium strategies after period m. Let s∗L and s∗F denote the equilibrium strategies of both firms. Then, the project values are formally written by (x (m−1),xF (m−1))

(m, y) = UL (m, s∗L , s∗F , xL (m − 1), xF (m − 1), y) = EQ [uL (m, s∗L , s∗F , xL (m − 1), xF (m − 1), Y m )|Y(m) = y], (xL (m−1),xF (m−1)) VF (m, y) = UF (m, s∗F , s∗L , xF (m − 1), xL (m − 1), y) = EQ [uF (m, s∗F , s∗L , xF (m − 1), xL (m − 1), Y m )|Y(m) = y].

VL L

We define that the cash flows obtained by both firms at matu(x (M−1),xF (M−1)) (M, y) = rity period M are supposed to be zero, i.e., VL L (xL (M−1),xF (M−1)) VF (M, y) = 0 for any demand y and any states (xL (M−1), xF (M− 1)). (x (m−1),xF (m−1)) (x (m−1),xF (m−1)) VL L (m, y) and VF L (m, y) are obtained by the fol(x (m),xF (m)) lowing recursive procedure for m = 0, · · · , M − 1. Let vi L (m, y) denote the project values (not including the investment cost) after the decisions of both firms at period m, assuming that both firms follow equilibrium strategies after period m + 1. (1,1) (1,1) First, we consider the values of VL (m, y) and VF (m, y) which are easy to compute because both of the firms have already invested and there are no decision to make at period m. Thus, the project values can be derived by adding the discounted expected values of the next period to the current cash flow: (1,1)

(3)

Vi

(1,1)

(m, y) = vi

(m, y)

for any i = L, F where (1,1)

vi

h (1,1) i (m, y) = D11 y∆t + e−r∆t EQ Vi (m + 1, Y(m + 1))|Y(m) = y . (1,0)

Next, consider the project value of VF (m, y). In this case firm L has already invested in the project and firm F can make an investment decision

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solely. If firm F decides to invest in the project at period m the project value (1,1) can be written as vF (m, y) − I. On the other hand, if firm F decides not to (1,0) invest at period m the project value is equal to vF (m, y) where h (1,0) i (1,0) vF (m, y) = D01 y∆t + e−r∆t EQ VF (m + 1, Y(m + 1))|Y(m) = y . Firm F makes the optimal decision to maximize its project value at period m. Thus, the following equation is satisfied. (1,0)

VF

(4)

(

=

(m, y) (1,1) (1,1) (1,0) vF (m, y) − I vF (m, y) − I > vF (m, y), (1,0) vF (m, y) otherwise. (1,0)

Next, we consider the project value of VL (m, y). Firm L which has already invested in the project has no decision to make at period m but the project value is dependent on the firm F’s decision. Thus, the following equations hold. (1,0)

VL

(5)

(

= (1,0)

where vL

(1,0)

vL

(m, y) (1,1) (1,1) (1,0) vL (m, y) vF (m, y) − I > vF (m, y), (1,0) vL (m, y) otherwise,

(m, y) is given by

h (1,0) i (m, y) = D10 y∆t + e−r∆t EQ VL (m + 1, Y(m + 1))|Y(m) = y .

Due to a symmetric aspect of the competition between firm L and firm F, the project values at state of (0, 1) are derived correspondingly: (0,1)

VL

(6)

(

=

(m, y) (1,1) (1,1) (0,1) vL (m, y) − I vL (m, y) − I > vL (m, y), (0,1) otherwise, vL (m, y)

where (0,1)

vL

h (0,1) i (m, y) = D01 y∆t + e−r∆t EQ VL (m + 1, Y(m + 1))|Y(m) = y ,

and (0,1)

(7)

VF =

(

(m, y) (1,1) (1,1) (0,1) vF (m, y) vL (m, y) − I > vL (m, y), (0,1) vF (m, y) otherwise,

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Invest

Not invest

(v

Invest (1,1)

L

(v

− I , vF − I

( 0,1)

L

(1,1)

,

( 0,1)

vF

−I

)

)

Not invest

(v

(1,0 )

L

(v

− I , vF1,0

( 0,0 )

L

(

,

( 0,0 )

vF

)

)

)

Figure 2. The payoff matrix in case of simultaneous decision.

where (0,1)

vF

h (0,1) i (m, y) = D10 y∆t + e−r∆t EQ VF (m + 1, Y(m + 1))|Y(m) = y .

Finally, the project values at state of (0, 0) are considered when both firms have options to invest in the project. In this case actions for both firms in the equilibrium strategies can be derived from a Nash equilibrium of one shot game described by the payoff matrix illustrated in Fig. 2. (0,0) In the figure, vL (m, y) is defined by (0,0)

vL

h (0,0) i (m, y) = D00 y∆t + e−r∆t EQ VL (m + 1, Y(m + 1))|Y(m) = y .

There could be multiple Nash equilibria in the payoff matrix in Fig. 2, hence we need to define additional criteria to choose the unique equilibrium. As explained in the introduction, we introduce two criteria which are called LAC and 50% criterion. In the LAC, we assume that firm L has a competitive advantage in the case of the multiple equilibria. In the payoff matrix shown in Fig. 2, (1,1) (1,1) (1,0) (0,1) (0,1) (1,0) equations vL = vF , vL = vF and vL = vF are satisfied at any period. Hence, we can determine the equilibrium strategies using these equations. Table 1 summarizes the equilibrium strategies that can occur under LAC. By using above equations, all possible cases are classified into nine cases. The second column labelled ”conditions of equilibria” in Table 1 corresponds to the conditions which determine equilibria. The resulting equilibria are shown in the third column denoted by the pair of the states, (sL (m, xL (m − 1), Y(m)), sF (m, xF (m − 1), Y(m))). For example, (1, 0) represents the state where only firm L finishes the investment. Note that the third and forth cases correspond to symmetric equilibria while the fifth and sixth cases correspond to asymmetric equilibria. If there exist multiple equilibria, one equilibrium strategy is selected by LAC, which is reflected as an inequality in the fourth column labelled ”condition of equilibrium selection”. In the fifth column, the selected equilibrium by LAC is shown. The last two columns correspond to the project values of firm L and firm F in the equilibrium, respectively.

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It is important to note that in the case of symmetric equilibria, which correspond to the third and forth cases, LAC is equivalent to another criterion for equilibrium selection called payoff dominance. In the payoff dominance criterion we choose the unique equilibrium if all players’ payoffs in the equilibrium are greater than those of the other equilibrium. After Harsanyi and Selten (1988) apply this criterion to their theory of equilibrium selection it is often used in game theory. Fudenberg and Tirole (1985) and Huisman (2001) also use this criterion to select an equilibrium. Note that firm L’s project value in the equilibrium selected by LAC is greater than any other value in the multiple equilibria by definition. In addition, firm F’s project also takes the maximum values in that equilibrium because of the symmetry of the strategies between two firms. Consequently, the equilibrium selected by LAC in symmetric equilibria is consistent to that chosen by payoff dominance criterion. In applying the 50% criterion, there are four possible cases which are (x ,x ) summarized in Eq. (8). By symmetric definition of our model vL L F = (x ,x ) (x ,x ) (x ,x ) vF L F and VL L F = VF L F hold for any state of (xL , xF ). Consequently, the (0,0) (0,0) project values of VL and VF are calculated by:

(8)

VL(0,0) (m, y) = VF(0,0) (m, y)  (1,1)  vL (m, y) − I         (1,1) (0,0)   21 (vL (m, y) − I + vL )      = 1 (1,0)   (v (m, y) − I + v(0,1) )  L 2 L          v(0,0) (m, y)    L

v(1,1) (m, y) − I > v(0,1) (m, y) L L and v(1,0) (m, y) − I > v(0,0) (m, y), L L (1,1) (0,1) vL (m, y) − I > vL (m, y) and v(1,0) (m, y) − I < v(0,0) (m, y), L L (1,1) vL (m, y) − I < v(0,1) (m, y) L and v(1,0) (m, y) − I > v(0,0) (m, y), L L (1,1) (0,1) vL (m, y) − I < vL (m, y) and v(1,0) (m, y) − I < v(0,0) (m, y). L L

The first case indicates that both firms choose to invest at period m when the previous state is (0, 0) and in the fourth case neither firm invests in the project at period m. There exists the unique equilibrium in these cases. On the other hand, the second and third case corresponds to multiple equilibria where the states of (1, 0) and (0, 1) satisfy the equilibrium condition. In these cases the project value can be given by the average of the two project values in the equilibria under 50% criterion. 3. Numerical experiences and economical implication 3.1 Selection of parameters In this section we analyze a relation between competition and flexibility under uncertainty. Especially, we focus on the effects of FMA and SMA on the project values as well as the equilibrium strategies. In the numerical

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(0,0)

Table 1 Calculation of VL

(0,0)

and VF

conditions of equilibria 1 v(1,1) L (1,1) v 2 L (1,0) v F (1,1) v L 3 (1,0) v F (1,1) v 4 L (1,0) v F (1,1) v 5 L (1,0) v F (1,1) v 6 L (1,0) v F (1,1) v L 7 (1,0) v F (1,1) v 8 L (1,0) v F (1,1) v 9 L (1,0) v F

(0,1) −I >v , L (0,1) , −I >v L (0,0) −I >v F (0,1) , −I >v L (0,0) −I v L (0,0) −I 1, there exists a positive constant Cp such that E[(1 + sup |Su | + sup |Gu |)p |St = S, Gt = G]

(2.4)

t 0 such that ψ(y, t) > 0 for y < y 0 (t) and ψ(y, t) < 0 for y > y0 (t). If r ≥ q, then ψ(y, t) < 0 holds for any y > 0. We set y 0 (t) = 0 in the latter case. Theorem 3.2. Consider the American average strike put option. The exercise region E is nonempty. For any t, its section Et := {y; (y, t) ∈ E} is equal to the upper half line [y∗a (t), ∞), where y0 (t) ≤ y∗a (t) and y∗a (t) > 0. Further, the value function W A (y, t) is decomposed as ∞ A −q(T−t) (3.7) W (y, t) = e max{eY/T − 1, 0}F(y, t, Y, T)dY +

−∞

T

e−q(u−t)

t

∞ y∗a (u)

(−q + reY/u +

Y Y/u e )F(y, t, Y, u)dYdu. u2

The first term of the right hand side coincides with WE (y, t), the value function of the average strike put option discussed in the previous section. The second term of the above is called the early exercise premium of the holder. It is positive because the integrand is positive. Proof. We first show that for any 0 < t < T inequality WE (y, t) < ξ(e y/t ) holds for sufficiently large y. A direct computaion yields ϕt,T (y)/T

(3.8) e

=e

y/T

T 1 σ2 T 2 − t2 T 2 − t2 1 exp − (r − q) uσdZ∗u − exp . 2 T T t 4 T

The last member exp

T 1 T

t

uσdZ∗u −

σ2 T2 −t2 4 T

with mean 1. Therefore, W E (y, t) ≤ E∗ [e−q(T−t) eϕt,T (y)/T ]

is a positive martingale

December 26, 2006

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180

T 2 − t2 1 ≤e e exp − (r − q) 2 T 1 ≤ e y/T exp − r(T − t). 2 y/T −q(T−t)

W E (y,t)

Consequently, if t < T, ey/t < 1 holds for sufficiently large y. This implies W E (y, t) < ξ(e y/t ) for sufficiently large y. Now since WA (y, t) ≥ ξ(e y/t ) holds for any y, t, we get W A (y, t) > W E (y, t) for any t, proving that Et is nonempty for any t and sufficiently large y. We want to show that C is connected, which ensures that Ct is an interval and Et is an upper half line [y∗ (t), ∞). We follow the argument of Oksendale [9]. Let U ⊂ C be the connected component including {(y, t); ψ(e y/t) > 0}. (If the above set is empty take any connected component in C) Then U = C holds. Indeed, if it is not connected there exist another connected component V in C. Let τ be the leaving time from the set V. Then for (y, t) ∈ V we have (L∗ (t) + ∂t∂ − q)ξ(e y/t ) = ψ(e y/t ) ≤ 0. Therefore E∗ [e−q(T∧τ−t) ξ(eϕt,T∧τ (y)/T∧τ )] T∧τ ∂ − q)ξ(eϕt,u (y)/u )du] e−q(u−t) (L∗ (u) + = ξ(e y/t ) + E∗ [ ∂u t ≤ ξ(e y/t ). The left hand side of the above is equal to W A (y, t). Therefore we have W A (y, t) ≤ ξ(e y/t ), which contradicts to (y, t) ∈ C. Now the section Ct is a one dimensional connected set (interval), which includes (−∞, y0(t)). Then Et is a upper half line [y∗a (t), ∞), where y0 (t) ≤ y∗a (t). Further, Et does not contain 0, so that y∗a (t) > 0. The function WA (y, t) is locally Lipschitz continuous with respect to (y, t). (Kunita-Seko [5]). For any t, the Radon-Nikodym derivative W A y (y, t) (with respect to y) is continuous with respect to y except possibly at the critical point y∗a (t). At the critical point y = y∗a (t), we have W A y (y−, t) = A W y (y+, t) by the contact property. Furthermore, for any t, W A y (y, t) is continuously differentiable with respect to y for y y∗ (t). Then W A y (y, t) is absolutely continuous with respect to y for any t. We can apply Ito’s ˆ formula (Theorem 6.4 in Appendix) to the value function WA (y, t). Then we have e−q(T−t) W A (ϕt,T (y), T) − W A (y, t) T e−q(u−t) W yA (ϕt,u (y), u)σudZ∗ (u) + =− t

T t

e−q(u−t) (L∗ (u) +

∂ − q)W A (ϕt,u (y), u)du. ∂u

December 26, 2006

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Asianput

181

Since W A (ϕt,T (y), T) = ξ(eϕt,T (y)/T ) holds, we get E

T

A

W (y, t) − W (y, t) = t

E∗ [e−q(u−t) ψ(ϕt,u (y), u)1 y(u)>y∗a (u) ]du.

Therefore the value function is written by (3.7). Finally the value function VA (S, G, t) can be computed through the formula V A (S, G, t) = SW A (t log GS , t). 4. Average Strike Put Options of the Game Type We shall consider the game type Asian option. The pay-off function of the holder of the option is given by f (S, G) and that of the writer is given by fδ (S, G) = f (S, G) + δ(S, G) (> f (S, G)), where δ(S, G) denotes the penalty payed by the writer in the case where he cancel the option. The price of the option is then given by V(S, G, t) := inf sup JS,G,t(σ, τ), σ∈Tt,T τ∈T t,T

where JS,G,t (σ, τ) := E0 [e−r(σ∧τ−t) { f (Sτ , Gτ )1τ≤σ + fδ (Sσ , Gσ )1σ δ for t < t∗δ . If W A (0, 0) < δ, then W A (0, t) < δ holds for all t. In this case we set t∗δ = 0. Theorem 4.1. Consider the average strike put option of the game type. Assume 0 ≤ q ≤ r. 1) EB is a nonempty closed set and the section EBt is equal to the upper half line [y∗δ (t), ∞), where 0 < y∗δ (t) ≤ y∗a (t). 2) The writer’s cancellation region EA is equal to E0 := {(0, t); t ≤ t ∗δ }, i.e., {0}, if t ≤ t∗δ , A (4.5) Et = φ, if t > t∗δ . The first assertion on the exercise region EB can be verified similarly as Theorem 3.2. In order to prove the second assertion, we prepare two lemmas. We consider the function (4.6)

˜ W(y, t) := sup J∗y,t (σ0 , τ), τ

˜ where σ0 is the hitting time to the set E0 . We want to prove W(y, t) = W(y, t). ∗ A ˜ Note first that W(y, t) = supτ J y,t (T, τ) = W (y, t) holds if t > δ∗a , where W A (y, t) is the value function of the American average strike put option. We consider the case where t < t∗a .

December 26, 2006

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Asianput

183

˜ Lemma 4.2. The function W(y, t) is nondecreasing with respect to y. Further for any y > 0 it is nonincreasing with respect to t. Proof. Sincce the pay-off functions ξ and ξδ are nondecreasing functions of y, the function J∗y,t (σ0 , τ) is also nondecreasing with respect to y. Then ˜ W(y, t) is also nondecreasing with respect to y. In order to prove the decreasing property with respect to t, we consider ˜ G, t) := sup JS,G,t(σ0 , τ). V(S,

(4.7)

τ∈Tt,T

It holds ˜ G, t) = sup E0 [e−r(τ∧σ0 −t) { f (Sτ∧σ0 , Gτ∧σ0 ) + δSσ0 1σ0 0, similarly as the case of American option. We want to ˜ show W(y, t) < ξδ (e y/t ) holds for any y > 0. Fixed t, consider the function ˜ ˜ U(y) := W(y, t) − ξ(e y/t ). It satisfies the boundary condition (terminal

December 26, 2006

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Asianput

184

˜ ∗) = U ˜ y ( y˜ ∗ ) = 0 since at the boundary point y = y˜ ∗ the condition) U(y ˜ y (y, t) = ∂ (ξ(e y/t )) holds valid. Further for y < y˜ ∗ (t) it contact property W ∂y satisfies the second order ordinary differential equation ˜ (L∗ (t) − q)U(y) = K(y),

(4.8) where

K(y) = −

˜ ∂W (y, t) − {q − re y/t }1{0 0 ≥ λ2 and eλ1 (y− y˜ ) and eλ2 (y− y˜ ) are fundamental solutions of the homogeneous equation (L∗ (t) − q)U = 0. The boundary conditions imply C1 = C2 = 0. Therefore we get ˜ U(y) =

1 λ1 − λ2

y y˜ ∗

K(z)(eλ1 (y−z) − eλ2 (y−z) )dz,

−∞ < y < y˜ ∗ .

˜ Then U(y) > 0 holds for 0 < y < y˜ ∗ . Further we have ˜ y (y) = U

1 λ1 − λ2

y y˜ ∗

K(z)(λ1 eλ1 (y−z) − λ2 eλ2 (y−z) )dz,

˜ is strictly decreasing. which is negative for any −∞ < y < y˜ ∗ . Therefore U ˜ ˜ ˜ t) < Since U(0) = δ, we have U(y) < δ for any 0 < y < y˜ ∗ , proving W(y, y/t ∗ ξδ (e ) for any 0 < y < y˜ . ˜ Next, if y < 0, we have always W(y, t) < δ = ξδ (e y/t ). The proof is complete. Proof of Theorem 4.1. Assume that t > t ∗δ . Then we have the inequalities W(y, t) ≤ W A (y, t) < ξδ (e y/t ) for any t. Therefore EA = φ and we have W(y, t) = W A (y, t) for any y. Assume next that t ≤ t ∗δ . The value function ˜ W(y, t) of the game option is less than or equal to W(y, t). Therefore the function W(y, t) satisfies the inequality W(y, t) < ξδ (e y/t ) for any y 0. This ∗ proves that EA t = {0} if t < δ . The proof is complete.

December 26, 2006

11:9


Asianput

185

5. Early Exercise Premium and Early Cancellation Fee Lemma 5.1. The value function of the average strike put option of game type satisfies ∂W (0−, t) = ∂y ∂W (0−, t) < ∂y

(5.1) (5.2)

∂W (0+, t), ∂y ∂W (0+, t), ∂y

if t ≥ t∗δ , if t < t∗δ .

Proof. Let τ∗ be the hitting time to the set EB and σ0 be the hitting time ˆ to the set E0 as before. Set W(y, t) := J ∗y,t (T, τ∗ ) and conisder the function ˆ ˆ ˆ t) = 0 U(y, t) := W(y, t) − W(y, t). It is a nonnegative function. If t ≥ t ∗δ , U(y, ˆ holds for any y. Then W(y, t) = W(y, t) is continuously differentiable at y = 0. Therefore we get (5.1). ˆ t) is We shall consider the case where t < t∗δ . Then the function U(y, ∗ ˆ is represented by positive for y < yδ (t). The function U ˆ ˆ t,σ0 (y), σ0) − ξδ (eϕt,σ0 (y)/σ0 )}1σ0 0. Further for any t, it is increasing with respect to y < 0. ˆ ˆ Now the function U(y) := U(y, t), 0 < y < y ∗δ is represented by (5.3)

∗ ∗ ˆ U(y) = C1 eλ1 (y−yδ ) + C2 eλ2 (y−yδ ) +

where L(z) = −

1 λ1 − λ2

y y∗δ

L(z)(eλ1 (y−z) − eλ2 (y−z) )dz,

ˆ ∂U (z, t) ≥ 0. ∂t

ˆ ∗ ) = 0 and Further it satisfies the boundary (terminal) condition U(y δ ∗ ˆ U y (yδ ) ≤ 0. Therefore we have C1 + C2 = 0 and C1 λ1 + C2 λ2 ≤ 0. This implies C1 ≤ 0 and C2 = −C1 . Then we have y 1 λ1 (y−y∗δ ) λ2 (y−y∗δ ) ˆ −e )C1 + L(z)(eλ1 (y−z) − eλ2 (y−z) )dz. U(y) = (e λ1 − λ2 y∗ δ

If C1 < 0, the first term of the right hand side is positive. If C1 = 0, the ˆ last term of the above is positive, since U(0) > 0. In the latter case the ineqquality L(z) > 0 holds on some interval.

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ˆ y is written by Now the derivative U ˆ y (y) = (λ1 eλ1 (y−y∗δ ) − λ2 eλ2 (y−y∗δ ) )C1 + U

1 λ1 − λ2

y y∗δ

L(z)(λ1 eλ1 (y−z) − λ2 eλ2 (y−z) )dz.

Both terms of the right hand side are nonpositive. Further if C1 < 0, the first term is negative and if C1 = 0, the second term is negative. This ˆ y (0−, t) ≥ 0, ˆ y (0+, t) < 0. On the other hand for y < 0, we have U implies U ˆ since U(y, t) is a nondecreasing function of y < 0. These two inequalities ˆ imply (5.2) because the function W(y, t) is continuously differentiable at y = 0. Theorem 5.2. Assume 0 ≤ q ≤ r. If t ≥ t∗δ , we have W(y, t) = W A (y, t), where WA (y, t) is the value function of the American average strike put option. If t < t∗δ , we have W(y, t) < W A (y, t). Further W(y, t) is decomposed as

(5.4)

W(y, t) = e−q(T−t)

T

∞

−∞

max{eY/T − 1, 0}F(y, t, Y, T)dY

∞

Y Y/u e )F(y, t, Y, u)dYdu u2 T ∂W σ2 −q(u−t) 2 ∂W e u (0+, u) − (0−, u) F(y, t, 0, u)du. − 2 t ∂y ∂y

+

e−q(u−t)

y∗δ (u)

t

(−q + reY/t +

Remark. The first term of (5.4) coincides with the value of the Europian option. The second term is called the early exercise premium of the holder and the third term is called the early cancellation fee of the writer. Proof. We apply Ito’s ˆ formula to the function W(y, t). It is locally Lipschitz continuous in (y, t). The Radon-Nikodyme derivative W y (y, t) (with respect to y) is a function of bounded variation with respect to y. It admits the Lebesgue decomposition: dW y (y, t) = W yy (y, t)dy + {W y (y+, t) − W y (y−, t)}δ0(dy), where δ0 is the delta function concentrated at y = 0. Then setting ϕu = ϕt,u we have by Theorem 6.4 in Appendix T −q(T−t) W(ϕT (y), T) = W(y, t) − e−q(u−t) W y (ϕu (y), u)σudZ∗(u) e

t T

∂ − q)W(ϕu (y), u)du e−(u−t) (L∗ (u) + ∂u t T ∂W −q(u−t) 2 2 ∂W (0+, u) − (0−, u) dL0u (y), + e σu ∂y ∂y t

+

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where L0u (y) is the local time of the flow ϕu (y) at the level 0. The expectation of the first term is equal to WE (y, t). Then we get (5.5)

W E (y, t) = W(y, t) + E∗ +E∗

t T

T

e−q(u−t) ψ(eϕu (y)/u )1{ϕu (y)>y∗δ (u)} du)

e−q(u−t) σ2 u2

t

∂W ∂W (0+, u) − (0−, u) dL0u (y) . ∂y ∂y

The second term of the right hand side is computed by T ∞ Y e−q(u−t) (q − reY/u − 2 eY/u )F(y, t, Y, u)dYdu. u t y∗δ (u) The last term is ∂W σ2 T −q(u−t) 2 ∂W (0+, u) − (0−, u) F(y, t, 0, u)du. e u 2 t ∂y ∂y Here we used the equality. E

∗

[L0v (y)]

1 = 2

v

F(y, t, 0, u)du. t

6. Appendix: A Generalized Ito’s ˆ Formula and Local Time of a One Dimensional Stochastic Flow. Consider a stochastic differential equation on Rd . (6.1)

dϕt =

m

v j (t, ϕt )dB j (t),

01

xN(ds, dx) + 0

|x|≤1

xN(ds, dx)

where N(dt, dx) = N(dt, dx) − dtν(dx). Here W is a Brownian motion and N is a Poisson random measure. We study a financial market where the price of the risky asset is given in terms of the exponential of X as St = S0 eXt (such a process is called “geometric Lévy process” or “exponential Lévy process”) and where the risk less asset has a constant interest rate r. Q For any probability measure Q equivalent to P, we denote by (Lt , t ≥ 0) 193

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Ritusumei2006˙miyahara

194

the strictly positive P-martingale such that dQ|Ft = LQ t dP|Ft . From the predictable representation theorem of positive martingales (see [3] ) there exists two predictable processes f and g such that Q Q g(t,x) − 1) N(dt, dx) . dLt = Lt− ft dWt + (e In what follows, we shall denote by L( f, g) such a density, and by Q( f,g) the corresponding equivalent martingale measure. The martingale property of ( St = St e−rt , t ≥ 0) under Q( f,g) holds if and only if for any t the equality (1) ft σ + (e g(t,x) (ex − 1) − x11|x|≤1 )ν(dx) = β IR

holds almost surely, where β = −(b + 12 σ2 − r) (see [3] ). We shall denote by C the set of pairs ( f, g) such that (1) holds and by C0 the set of nonrandom pairs such that (1) holds (in that case, f is a real number and g a real function of the real variable x). Our aim is to find the density L, i.e., a pair of predictable processes ( f ∗ , g∗ ) such that E(L2T ( f ∗ , g∗ )) = inf{E(L2T ( f, g)) , ( f, g) ∈ C }. The solution of this problem is the “minimal variance martingale measure” (MVMM). We denote it by P (MV) . Remark 1.1. M. Schweizer [4] introduced the “variance optimal martingale measure” (VOMM). The VOMM is a signed measure in general. In the case where the VOMM is positive, the VOMM is identified with MVMM by definition. We prove that, under some conditions, there exists a non random optimal pair ( f ∗ , g∗ ). Theorem 1.1. Assume that the following equation for ( f, g(x), µ) f = µσ (e µσ2 +

g(x)

− 1) = µ(ex − 1)

1 + µ(ex − 1) (ex − 1) − x11|x|≤1 ν(dx) = β

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195

has a solution ( f ∗ , g∗ (x), µ∗). Then it holds that ∗ ∗ (i) The martingale measure Q( f ,g ) is the MVMM for the process St . (ii) The Lévy measure of (Xt , t ≥ 0) under P(MV) is ∗ ν(MV) (dx) = e g (x) ν(dx) = 1 + µ∗ (ex − 1) ν(dx) When the conditions of the previous theorem are not satisfied, we don’t know whether the MVMM exists or not. We define an “-optimal” MVMM, Q = Q( f ,g ) , as a martingale measure which satisfies the following condition E[L2 ( f , g )] ≤ E[L2 ( f, g)] + , ∀( f, g) ∈ C. Theorem 1.2. For any > 0, there exists an -optimal MVMM in the class of non-random controls. Theorem 2 suggests us that the following conjecture may hold true. Conjecture: If the MVMM exists, then it is in the class C 0 . We can generalize the above problem to the “minimal Lq martingale q measure” (MLq MM), q > 1, and we denote it by P(ML ) . The problem is to find predictable processes f ∗ and g∗ such that E{(LT ( f ∗ , g∗ ))q } = inf E{(LT ( f, g))q; ( f, g) ∈ C}. We obtain the following theorem, Theorem 1.3. Assume that the following equation for ( f q , gq (x), µq ) q(q − 1) fq = µq σ (q−1)gq (x)

q(e

µq σ2q q(q − 1)

+

(1 +

− 1) = µq (ex − 1)

µq (ex − 1) q

1 ) q−1 (ex − 1) − x11|x|≤1 ν(dx) = β.

has a solution ( fq∗ , g∗q (x), µ∗q ). Then ∗ ∗ (i) The martingale measure Q( fq ,gq ) is the minimal Lq martingale measure for the process St . q (ii) The Lévy measure of Xt under P(ML ) is (MLq )

ν

(dx) = e

g∗q (x)

1 q−1 µ∗q x ν(dx) = 1 + (e − 1) ν(dx) q

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196 q

Next we consider the limit limq↓1 P(ML ) . q

Theorem 1.4. Assume that the ML q MM P(ML ) , 1 < q ≤ 2, and the minimal entropy martingale measure P(MEMM) exist, then under some additional assumptions it holds that q ∗ x lim ν(ML ) (dx) = eθ (e −1) ν(dx) = ν(MEMM) (dx). q↓1

q

This result means that “when q ↓ 1, minimal q martingale measure P(ML ) converges to the minimal entropy martingale measure P(MEMM) in some sense.” (See [1] for the Lévy measure of X t under P(MEMM) .) 2. Remarks The similar problem as ours have been discussed by many persons. But almost all papers are studied under the frame work of signed martingale measure or only absolute continuity. On the other hand, our discussions are in the frame work of the equivalent martingale measures. So our results are obtained in the set up of the equivalent martingale measures. The similar results with Theorem 1.4 are obtained for continuous processes ([2], etc.). Our result is an extension of such results to the case of jump processes. References 1. Fujiwara, T. and Miyahara, Y. (2003), The Minimal Entropy Martingale Measures for Geometric Lévy Processes, Finance and Stochastics 7, 509–531. 2. Grandits, P. and Rheinländer, T. (2002), On the minimal entropy martingale measure, The Annals of Probability, 30(3), 1003–1038. 3. Kunita, H. (2004), Representation of Martingales with Jumps and Applications to Mathematical Finance, Stochastic Analysis and Related Topics in Kyoto, In honour of Kiyosi Itô, Advanced Studies in Pure Mathematics 41, Mathematical Society of Japan. 4. Schweizer, M. (1996), Approximation Pricing and the Variance-Optimal Martingale Measure, Annals of Probability 24, 206–236.

December 26, 2006

11:13


litterer-lyons-updated

Cubature on Wiener Space Continued Christian Litterer and Terry Lyons Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

Higher order particle methods can provide an accurate description of an evolving family of measures, but frequently the number of particles used in the description explodes as we iterate. In this paper we present a general method to simplify the support of the intermediate measures used in the iteration without increasing the error in the particle approximation by more than a constant factor. We describe two algorithms that can be used to simplify the support of a discrete measure and give an application to the cubature on Wiener space method developed by Lyons, Victoir [13]. Key words: particle systems, stochastic analysis, cubature, stochastic filtering 1. Introduction Probabilistic understandings are producing novel numerical techniques for integrating partial differential equations of parabolic type. For example see the work of Kusuoka [9]. We consider a Stratonovich stochastic differential equation (1)

dξt,x = V0 (ξt,x )dt +

d

Vi (ξt,x ) ◦ dBit , ξ0,x = x

i=1

defined by a family of smooth vector fields Vi and driven by Brownian motion. It is well known that computing a weak solution to (1) and setting PT−t f := E( f (ξT−t,x )) solves a parabolic partial differential equations (PDE). More precisely u(t, x) = E( f (ξ T−t,x )) solves

(2)

∂u (t, x) = −Lu(t, x), ∂t u(T, x) = f (x) 197

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198

where L is the differential operator given by 1 L = V0 + (V12 + . . . Vd2 ). 2 Kusuoka’s algorithm [9] and the Kusuoka-Lyons-Victoir (or KLV) method [13] are two higher order methods for weak approximations of SDEs. In both approximation schemes the number of calculations grows exponentially in the number of iterations; nonetheless the methods can be highly effective in practice. In [15] Ninomiya and Victoir present an application to pricing Asian options under a Heston stochastic volatility model. They evaluate an algorithm, that might (non-trivially) be interpreted as a special case of the KLV method of degree five (in combination with Romberg extrapolation and quasi Monte-Carlo) for the SDE arising in this case and obtain a speed up by a factor of 10 6 compared to the classical Euler-Maruyama scheme. Similar results are obtained by Ninomiya [16] and Ninomiya, Kusuoka [10] for Kusuoka’s algorithm. They combine the method with the tree based branching algorithm (TBBA), a partial sampling scheme developed by Crisan, Lyons [3], and apply it to price European options under a Heston model. In this paper we give a general methodology for recombining particles in a particle systems by reducing the support of the discrete measure. We describe two algorithms that may be used to compute such reduced measures and we announce results of an application to the KLV method. We can show that if we interpret the combined method as a particle system, then under the uniform Hormander ¨ condition for the vector fields Vi the number of particles grows polynomially in the number of iterations. Finally we outline an application of the recombination methods to the stochastic filtering problem. 2. Higher order methods for weak approximations of SDEs In [9] Kusuoka develops a higher order method for weakly approximating the solution of stochastic differential equations (1). For the application of the method the test function f is required to be Lipschitz and vector fields must satisfy the so called UFG condition, which is weaker than the uniform Hormander ¨ condition (see section 6 for details). The author introduces the concept of a m-moment similar family of random variables. Let k A = {∅} ∪ ∞ k=1 {0, . . . , d} and α = (α1 , . . . , αk ) ∈ A be a multi-index. To take account of the special role of the vector field V0 in the direction of the drift of the SDE we define a weight on a multi-index α by α = k + card(j : α j = 0). Let A(j) = {α ∈ A : α ≤ j}.

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199

Define iterated integrals by B◦ α (t) =

0 supz {C(x, z) + u(z)} −

β 2

≥ supz {−C0 + C(x, ξ) + C(ξ, z) + β + u(z)} − = −C0 + C(x, ξ) + supz {C(ξ, z) + u(z)} + = C(x, ξ) + Mu(ξ) +

β 2

β 2

and we see that u(ξ) > Mu(ξ) +

(2.12) and so ξ ∈ C. Set

β 2

τˆ 1 = inf{t; u(Xt ) = M(Xt )}

and take ξˆ1 such that sup{C(Xτˆ 1 − , x ) + u(x )} − x

β < C(Xτˆ k − , ξˆ1 ) + u(ξˆ1 ). 2

Successively we define (n−1)

τˆ n = inf{t ≥ τˆ n−1 ; u(X¯ t

(n−1)

) = M(X¯ t

)}

β 2

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and take ξˆn such that β (n−1) < C(X¯ τˆ n−1 − , ξˆn ) + u(ξˆn ), 2n

(n−1)

sup{C(X¯ τˆ n−1 − , x ) + u(x )} − x

n = 2, 3, . . .. Theorem 2.1. Suppose that (2.6) has a solution (u, l) such that u ∈ W 1,∞ and assume the assumptions (2.2)-(2.4) and (2.10). Then the above defined straregy {(ˆτn , ξˆn )}n forms an optimal staretegy: sup J = sup lim sup

{τn ,ξn }n

{τn ,ξn }n

T→∞

1 ¯ = l. log E[exp{ΦT (X)}] T

and as a consequence the constant l of the solution to (2.6) is unique. Proof of Theorem 2.1. By construction of X¯ t , X¯ τˆ n = ξˆn and we have u(X¯ τˆ n − ) − u(X¯ τˆ n ) − {−C0 + C(X¯ τˆ n − , X¯ τˆ n ) +

β } u(X¯ T ) − u(X¯ τˆ n ) + − =

T 0

= − 12

Lu(X¯ s )ds +

T 0

T 0

i=1 {−C0

T

(∇u)σσ∗ ∇u(X¯ s )ds +

0

n

¯

i=1 {u(Xτˆ i − )

n

(∇u)∗ σdWs −

−

n

in C

β + C(X¯ τˆ i − , X¯ τˆ i ) + 2i }

n

i=1 {−C0

+ C(X¯ τˆ i − , X¯ τˆ i ) +

(∇u)∗ σdWs −

i=1 {−C0

− u(X¯ τˆ i−1 )}

T 0

( f (X¯ s ) − l)ds

β + C(X¯ τˆ i − , X¯ τˆ i ) + 2i }

Accordingly

T

e0

( f (X¯ s )−l)ds+ τˆ i ≤T {−C0 +C(X¯ τˆ i − ,X¯ τˆ i )}

¯

¯

1

≥ e−β−u(XT )+u(X0 )− 2

T 0

(∇u)σσ∗ ∇u(X¯ s )ds+

T 0

(∇u)∗ σdWs

.

β } 2i

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Thus we see that lim sup T→∞

T 1 ¯ ¯ ¯ log E[e 0 f (Xs )ds+ τˆ i ≤T {−C0 +C(Xτˆ i − ,Xτˆ i )} ] ≥ l T

On the other hand, u(y) ≥ Mu(y) and u(X¯ τi − ) ≥ −C0 + supz {C(X¯ τi − , z) + u(z)} ≥ −C0 + C(X¯ τi − , X¯ τi ) + u(X¯ τi ) for general strategy (τi , ξi )i , i = 1, 2, . . .. Therefore u(X¯ T ) − u(X¯ 0 ) ≤ u(X¯ T ) − u(X¯ τn ) + ni=1 {u(X¯ τi − ) − u(X¯ τi−1 )} −

n

i=1 {−C0

+ C(X¯ τi − , X¯ τi )}

and we have

T

E[e 0

f (X¯ s )ds+

T

≤ E[e 0

¯

¯

τi ≤T {−C0 +C(Xτi − ,Xτi )

]

f (X¯ s )ds−(u(X¯ T )−u(X¯ 0 ))+u(X¯ T )−u(X¯ τn )+ ni=1 {u(X¯ τi − )−u(X¯ τi−1 )}

.

Define a probability measure Pˆ by

T

1 T dPˆ ∗ ∗ ∗ | = e− 0 (∇ν) σdWs − 2 0 (∇ν) σσ ∇ν(Xs )ds . dP

ˆ t = Wt + t σ∇ν(Xs )ds is a Brownian Then under the probability measure W 0 motion process and so ˆ t. dXt = σ(Xt )dW ˆ = Since Xt is a symmeric diffusion process with the generator Lu 1 m ij ˆ D (a (x)u) under P, u(X ) − u(X ) admits a decomposition t 0 i, j=1 i 2

u(Xt ) − u(X0 ) =

t 0

ˆ s + Nt , (∇u)∗ σ(Xs )dW

ˆ ([6]). where Nt is an additive functional of zero energy corresponding to Lu Although the decomposition usually admits exceptional sets, they could be removed under the present setting because the relevant diffusion process has the transition density. Since we have ˆ + (∇ν)∗ a(x)∇u + 1 (∇u)∗ a∇u + f (x) − l ≤ 0 Lu 2

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in distribution sense t 1 {(∇ν)∗ a∇u(Xs ) + (∇u)∗ a∇u(Xs ) + f (Xs ) − l}ds) −(Nt + 2 0 is a positive additive functional. Therefore u(Xt ) − u(X0 ) ≤

t 0

−

ˆs (∇u)∗ σ(Xs )dW

t 0

{(∇ν)∗ a∇u(Xs ) + 12 (∇u)∗ a∇u(Xs ) + f (Xs ) − l}ds

By considering controlled process under the probability measure Pˆ we see that

T

E[e 0


T

ˆ 0 = E[e


T

×e 0

ˆ s− 1 (∇ν)∗ σ(X¯ s )dW 2

T

ˆ lT−(u(X¯ T −u(X¯ 0 ))+ 0 ≤ E[e

T

×e 0 ¯

¯

1

0

ˆ s− (∇u)∗ σ(X¯ s )dW

ˆ s− 1 (∇ν)∗ σ(X¯ s )dW 2

= E[elT−u(XT )+u(X0 )− 2

T

T 0

T

T 0

(∇ν)∗ a∇ν(X¯ s )ds

0

{(∇ν)∗ a∇u(X¯ s )+ 12 (∇u)∗ a∇u(Xs )}ds

(∇ν)∗ a∇ν(X¯ s )ds

(∇u)∗ a∇u(X¯ s )ds+

]

T 0

]

(∇u)∗ σ(X¯ s )dWs

]

Thus we see that lim sup T→∞

T 1 ¯ ¯ ¯ log E[e 0 f (Xs )ds+ τi ≤T {−C0 +C(Xτi − ,Xτi )} ] ≤ l T

because of u ∈ W 1,∞ and conclude our present theorem. References 1. C. Atkinson and P. Wilmott, Portfolio management with transaction costs: an asymptotic analysis of Morton and Pliska model, Mathmatical Finance 5(1995) 357–367. 2. A. Bensoussan and J. L. Lions Applications des Inéquations Variationnelles en Contrôle Stochastique, Dunod, Paris, 1978. 3. A. Bensoussan and J. L. Lions Contrôle Impulsionnel et Inéquations Quasi Variationneles, Dunod, Paris, 1982.

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4. T. R. Bielecki and S. R. Pliska, Risk sensitive asset management with transaction costs, Finance and Stochastics 4 (2000) 1–33. 5. A. Cadenillas, Consumption-investment problems with transaction costs: Survey and open problem, Math. Meth. Oper. Res 51 (2000) 43–68. 6. M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter, 1994. 7. A. J. Morton and S. Pliska, Optimal Portfolio Management with Fixed Transaction Costs, Math. Finance 5 (1995) 337–356. 8. H. Nagai, Risky Fraction Processes and Problems with Transaction Costs, Proceedings of the International Syposium on ”Stochastic Processes and Applications to Mathematical Finance”, eds. J Akahori, S. Ogawa, and S. Watanabe, World Scientific, (2004) 271–288. 9. H. Nagai, Stopping problems of certain multiplicative functionals and optimal investment with transaction costs, to appear in Appl. Mat. Optim. 10. H. Nagai, Risk-sensitive quasi-variational inequalities for optimal investment with general transaction costs, Asymptotic Analysis 48 (2006) 243–265. 11. S. Pliska and M. J. P. Selby, On a free boundary problem that arises in portfolio management, Phil. Trans. Royal Soc. London 347 (1994) 555–561. 12. M. Robin, On some impulse control problems with long run average cost, SIAM J. Control Optim. 19 (1981) 333–358. 13. T. Tamura, Maximizing growth rate of a portfolio with fixed and proportional transaction costs, to appear in Appl. Math. Optim. (2006)

December 26, 2006

13:43


levythu

A Convolution Approach to Multivariate Bessel Proceses Thu Van Nguyen1 , S. Ogawa2 , and M. Yamazato3 1

Department of Mathematics, International University, HCM City 2 Department of Mathematical Sciences, Ritsumeikan University 3 Department of Mathematics, Ryukyu University

In this paper we introduce and study Bessel processes {Bxt } which take values in a d-dimensional nonnegative cone R+d of Rd and are constructed via the multi-dimensional Kingman convolution . We prove that every d-variate Bessel process is a stationary independent increments-type process. Moreover, a stochastic integral with respect to {Bxt } with the convergence in distribution is defined. AMS 2000 subject classification: Primary 60G48, 60G51, 60G57; Secondary60J25, 60J60, 60J99 1. Introduction and Prelimilaries Let P denote the class of all p.m.’s on the positive half-line R+ endowed with the weak convergence and ◦ := ∗1,β denote the Kingman convolution (Hankel transforms ) which was introduced by Kingman [5] in connection with the addition of independent spherically symmetric random vectors in Eucliean n-space. Namely, for each continuous bounded function f on R+ we write : ∞ ∞ ∞ 1 Γ(s + 1) (1) f (x)µ ∗1,β ν(dx) = √ πΓ(s + 12 ) 0 0 0 −1 f ((x2 + 2uxy + y2 )1/2 )(1 − u2 )s−1/2 µ(dx)ν(dy)du, where µ, ν ∈ P , β = 2(s + 1) 1 (cf. Kingman [5] and Urbanik [15]. The Kingman convolution algebra (P, ◦) is the most important example of Urbanik convolution algebras (cf Urbanik [15]. In the language of Urbanik convolution algebras, the characteristic measure, say σs , of the Kingman convolution has the Rayleigh density (2)

σs (x) =

2 (s + 1)s+1 x2s+1 exp(−(s + 1)x2 )dx Γ(s + 1) 233

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levythu

234

with the characteristic exponent κ = 2 and the kernel Λs Λs = Γ(s + 1)Js (x)/(1/2x) s.

(3)

The radial characteristic function (rad.ch.f.) of a p.m. µ ∈ P, denoted by µ(u), ˆ is defined by ∞ Λs (ux)µ(dx), (4) µ(u) ˆ = 0 +

for every u ∈ R . In particular, the rad.ch.f. of σ s is σˆ s (u) = exp(−u2 ), u ∈ R+ .

(5)

It should be noted that, since the rad.ch.f. is defined uniquely up to the delation mapping x → ax, a > 0, x ∈ R+ , the representation (5) of the rad.ch.f. of σs may differ from that in Urbanik [15]. It is known (cf. Kingman [5], Theorem 1), that the kernel Λs itself is an ordinary ch.f. of a p.m., say Gs , defined on the interval [-1,1] as the following (6)

dGs (λ) =

Γ(s + 1) 1 2

1 2)

1

(1 − λ2 )s− 2 dλ

π Γ(s + 1 G− 1 = (δ1 + δ−1 ) 2 2 G∞ = δ0

1 (s ∈ (− , ∞)) 2 1 (s = − ), 2 (s = ∞).

Thus if θs denotes a r.v. with distribution Fs then for each t ∈ R+ , 1 (7) Λs (t) = Eexp(itθ s) = exp(itx)dG s(x). −1

Now we quote a definition of a Bessel process in Revus-Yor [12] Definition 1.1. A Bessel process is the square root of the following unique strong solution of the SDE t Zs dβs + βt, (8) Zt = x + 2 0

for any β 0 and x 0. It should be noted that Shiga and Watanabe [14] characterized the Bessel family as one-parameter semigroups of distributions on path spaces W = C(R+ , R) which stands for a convolution approach to Bessel processes. Our aim in this paper is to study Bessel processes via the Kingman convolution method. Therefore we will assume that the dimension β 1. Moreover, we will consider the Bessel process started at 0 only and will denote it by B(t), t 0.

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2. The Cartersian Product of Kingman Convolution Algebras This concept was introduced in Nguyen [10]. Namely, let P(Rk+ ) denote the class of all p.m.’s on Rk+ equipped with the weak convergence. Let F1 , F2 ∈ P(Rk+ ) be of the product form Fi = τ1i × ... × τki

(9) j

where τi ∈ P, j=1,2,... and i=1,2. We put F1 k F2 = (F11 ◦ F21 ) × ... × (Fk1 ◦ Fk2 ).

(10)

Since convex combinations of p.m.’s of the form (9) are dense in P(Rk+ ) the relation (10) can be extended to arbitrary p.m.’s on P(R k+ ). For every F ∈ P(Rk+ ) the k-dimensional radial ch.f. Fˆ is defined by ˆ = F(t)

(11)

k Rk+ j=1

Λs (t j x j )F(dx),

Let λ, λ1 , ..., λk be i.i.d. r.v’s with the common distribution Gs . Let X = (X1 , ..., Xk) be a Rk+ -valued random vector with distribution F. Further, suppose that r.v’s X and Λ, where Λ = (λ1 , ..., λk ), are independent. Set ΛX = {λ1 X1 , ..., λk Xk }

(12) and

d

Gs F = ΛX.

(13) Then, we have

F(y) = E(ei ),

(14) where y = (y1 , ..., yk) ∈ Rk+ In fact, we have E(e

and

i)

denotes the inner product in Rk .

=

Rk+

E(ei

k

j=1 (y j x j λ j

F(dx)

=

Rk+

Πkj=1 Λs (t j x j )F(dx)

= F(y). Thus, F(y) is an ordinary symmetric k-dimensional ch.f., and hence it is uniformly continuous. The following theorem is a simple consequence of (1.3) and (2.2).

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Theorem 2.1. The pair (P(R k+ , k ) is a commutative topological semigroup with δ0 as the unit element. Moreover, the operation k is distributive w.r.t. convex combinations of p.m.’s ∈ P(Rk+ ). In the sequel, the pair (P(Rk+ , k ) will be called a k-dimensional Kingman convolution algebra. For each vector x ∈ Rk+ the generalized translation operators (shortly, g.t.o.’s) Tx , x ∈ Rk+ acting on the Banach space Cb (Rk+ ) of real bounded continuous functions f are defined, for each y ∈ Rk+ , by x (15) T f (y) = f (u)δx k δy (du). Rk+

In terms of these g.t.o.’s the k-dimentional rad. ch.f. of p.m.’s on Rk+ can be characterized as the following: Theorem 2.2. A real bounded continuous function f on Rk+ is a rad.ch.f. of a p.m., if and only if f (0) = 1 and f is {T x }-nonnegative definite in the sense that for any x1 , ..., xk ∈ Rk and λ1 , ..., λk ∈ C k

(16)

λi λ j Txi f (x j ) 0.

i, j=1

(See [10] for the proof). Lemma 2.1. Every p.m. F defined on R k+ is uniquely determined by its rad.ch.f. F and the following formula holds: k F2 (t) = F1 (y)F2 (t), F1

(17)

where F1 , F2 ∈ P(Rk+ ) and y ∈ Rk+ . Proof. The formula (17) follows from formulas (1, 3, 12, 14). Now using the formulas (2, 3, 9) and a theorem of Weber ([6], p. 394) and integrating ˆ 1 u1 , ..., tk uk ), the function F(t t j , u j ∈ R+ , j = 1, ..., k, k − times w.r.t. σs , we get ˆ 1 u1 , ..., tkuk )σs (du1 )...σs (duk ) = (18) F(t Rk+

R+

...

k

R+ j=1

Λs (t j x j u j )F(dx)σs (du1 )...σs(duk ) =

k Rk+ j=1

exp{−t2j x2j }F(dx),

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which, by change of variables y j = x2j , j = 1, ..., k and by the uniqueness of the k-dimensional Laplace transform, implies that F is uniquely determined by the left-hand side of (18). Definition 2.1. A distribution F on Rk+ is said to be Rayleigh, if the Gs F defined by (12) is a k-dimensional symmetric Gaussian p.m. The following theorem is obvious and its proof is omitted. Theorem 2.3. A distribution F of a r.v. X on Rk+ is Rayleigh, if and only if for every x ∈ Rk+ the r.v. < x, X > is one-dimensional Rayleigh. It is the same as in the case k=1, the i.d. elements can be defined as the following: Definition 2.2. A p.m. µ ∈ P(R k+ is called i.d.if for every natural m there exists a p.m. µm such that µ = µm k ... k µm , (m − terms).

(19)

Moreover, a nonnegative stochastic process ξt , t ∈ T is said to be i.d., if each its finite dimensional distribution is i.d. Let ID(k ) denote the class of all i.d. elements in (P(Rk+ , k ). The following theorem is a slight generalization of Theorem 7 in Kingman [5]. Theorem 2.4. µ ∈ ID( k ) if and only if there exist a finite measure M on Rk+ with the property that M({0}) = 0 and for each y = (t 1 , ..., tk) ∈ Rk+ (20)

−logµ(y) ˆ =

Rk+

(1 −

k

Λs (< t j x j >)

j=1

1 + x2 M(dx), x2

where the integrand on the right-hand side of (21) is assumed to be (21)

lim (1 −

x→0

k

1 + x2 2 = tj . x2 j=1 k

Λs (< t j x j >)

j=1

In particular, if M = 0 then µ becomes a Rayleigh measure with the rad.ch.f. (22)

−logµ(y) ˆ =

k j=1

for any y ∈ Rk+

and λ j 0, j = 1, ..., k.

λ j t2j ,

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Proof. The proof of the first part of Theorem is a similar to that of Theorem 7 in Kingman [5]. To prove the remainder part we assume that k=2. The proof for the case k 3 is similar. For t, x ∈ R2+ we put H = H(t 1 , t2 , x1 , x2 ) :=

(23)

1 − Λs (t1 x1 )Λ(t2 x2 ) x21 + x22

By virtue of Kingman ([5], Formula (24)) and by the series representation of Λs (.) (Kingman [5], Formula (4))and by the fact that the measure G s is symmetric on the interval [−1, 1] we have 1 − Λs (t1 x1 )Λs (t2 x2 ) =

1 −1

{1 − Λs ((t21 x21 + t22 x22 + 2ut1 t2 x1 x2 )1/2 )}dGs (u)

= =

(24)

1

1 ( (t21 x21 + t22 x22 + 2ut1 t2 x1 x2 )dGs (u) − R −1 2

1 2 2 2 2 (t x + t x ) − R 2 1 1 2 2

where R is given by R=

∞ 1 −1

1 (− )r (t21 x21 + t22 x22 + 2ut1 t2 x1 x2 )r s!/r!(s + r)!dGs (u). 2 r=2

which implies that for fixed t1 , t2 we have (25)

lim

(x21 +x22 )→0

R = 0. x21 + x22

Consequently, such that for any t1 , t2 0 (26)

lim

(x21 +x22 )→0

1 − Λs (t1 x1 )Λs (t2 x2 ) = t22 + t22 x21 + x22

which proves (21). Now, letting M in (20) tend to measure zero and in2 tegrating both sides of (22) w.r.t. 1+x M(dx) we conclude, by virtue of x2 (25)and (26), that the formula (20) holds. Finally, since every projection of the limit p.m. is Rayleigh, it follows from Theorem Theorem 7 in Kingman [5]that the limit p.m. with rad.ch.f. of the form (22) must be a k-dimensional Rayleigh p.m. It is evident, from (22), that µ is Rayleigh in Rk+ if and only if for each y ∈ Rk+ the image of µ under the projection Πy x =< x, y > from Rk+ onto

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R+ is Rayleigh on R+ . Hence and by the Cramér property of the Kingman convolution (cf. Urbanik [16]) we have the following theorem: Theorem 2.5. Suppose that µ, ν ∈ P(R k+ ) and µ k ν is Rayleigh. Then both of them are Rayleigh. 3. Multivariate Symmetric Random Walks Given a p.m. µ ∈ P and n=1,2,... we put, for any x ∈ R + B(R+ ), the Borel σ-field of R+ , (27)

and B ∈

Pn (x, E) = δx ◦ µ◦n (E),

here the power is taken in the convolution ◦ sense. Using the rad.ch.f. one can show that {Pn (x, E)} satisfies the Chapman-Kolmogorov equation and therefore, there exists a homogenuous Markov sequence, say {Sxn }, n = 0, 1, 2,..., with {Pn (x, E)} as its transition probability. More generaly, we have Lemma 3.1. Suppose that {µ k , k = 1, 2, ...} is a sequence of p.m’s on Rk+ . For any 0 n < m, x ∈ Rk+ , E ∈ B(Rk+ ), (28)

Pn,m (x, E) = δx k µn k µn+1 k ... k µm−1 (E).

Then, {Pn,m(x, E)} satisfies the Chapman-Kolmogorov equation and therefore there exists a Markov sequence {Xxn }, n = 0, 1, 2, ... with Pn,m (x, E) as its transition probability. Proof. It can be proved by using the rad.ch.f. Since σs is i.d. w.r.t. the Kingman convolution the family of p.m.’s q(t, x, E) := σ◦t s ◦ δx (E) where t, x ∈ R, E is a Borel subset of R+ and the power is taken in the Kingman convolution sense, satisfies the Chapman-Kolmogorov equation and stands for a transition probability of a homogenuous Markov process Bxt , t, x ∈ R+ , such that , with probability 1, its realizations are continuous (cf. Nguyen [8] and Shiga-Wantanabe [14]). Let Hs be a k-dimensional Rayleigh measure with rad.ch.f. (20) and (29)

P(t, x, E) := H st s δx (E),

where t 0, x ∈ Rk+ , E is a Borel subset of Rk+ and the power is taken in the sense of convolution s . Then there exists a homogeneous Markov process, denoted by {Bxt } with values in Rk+ and transition probability (28).

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Definition 3.1. Every Markov process {B xt } with transition probability given by (28) is called a k-dimensional Bessel process. From the above definition and by (28) we have: Theorem 3.1. The rad.ch.f. of {B xt }, t 0 is of the form −logEΛ(< y, Bxt >=< y, x > t + t

(30)

k

λ2j y2j ,

j=1

where y ∈ Rk+ , λ j 0, j = 1, ..., k and t 0. j

j

j

Suppose that X j = {X1 , X2 , ..., Xk }, j = 1, 2 are Rk+ -valued independent r.v.’s with the corresponding distributions F j , j = 1, 2. Put X2 = {X11 ⊕ X12 , ..., Xk1 ⊕ Xk2 }. (31) X1 Then we get a k-dimensional radial sum of r.v.’s. By induction one can define such an operation for a finite number of r.v.’s. It is evident that the radial sum is defined up to distribution of r.v.’s and that the operation is associative. It is a natural problem to consider the usual multiplication of a Rk+ -valued r.v. and a nonnegative scalar. It is easy to see that the multiplication is distributive w.r.t. the radial sums defined by (31) which helps us to introduce the following stochastic integral. Definition 3.2. Let C be a σ-ring of subsets of a set X. A function M : C → L+ := L+ (Ω, F , P),

(32)

where L+ denotes the class of all nonnegative r.v.’s on (Ω, F , P), is said to be an k -scattered random measure, if • (i) M(∅) = 0 (P.1), • (ii) For any dent and

A, B ∈ C, A ∩ B = ∅, then M(A)andM(B) are indepen d M(B) M(A ∪ B) = M(A)

• (iii) For any A1 , A2 , ... ∈ C, pendent and (33)

the r.v.’s M(A j ), j = 1, 2, ... are inde-

d M(∪∞ j=1 A j ) =

∞ j=1

M(A j ),

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where the series on the right-hand side of (33) is convergent in distribution. It should be noted that the above definition of k -scattered random measure is subject to the equality in probability which, however, can be modified in the same way as Rajput and Rosinski ([11], Lemma 5.1 and Theorem 5.2) so that the new k -scattered random measure is defined almost surely. Specificly, we state without proof the above mentioned Lemma used by Rajput and Rosinski. Lemma 3.2. (O. Kallenberg) Let ξ and η be random elements defined on the probability space (Ω, P) and (Ω , P ), and taking values in the spaces S and T, respectively, where S is a separable metric space and T is a Polish space. d

Assume that ξ = f (η ) for some Borel measurable function f : T → S. Then d

there exists a random element η = η on the (”randomized”) probability space (Ω × [0, 1], P × Leb) such that η = f (η ) a.s. P × Leb. It is well known that if {W(t)}, t ∈ R+ is a Wiener process, then there exists a Gaussian stochastic measure N(A), A ∈ B0 , where B0 is the σ−ring of bounded Borel subsets of R+ with the property that, for every t 0, W(t) = N((0, t]). The same it is also true for Bessel processes. Namely, we get Theorem 3.2. Suppose that {B 0t } is a Bessel process started at 0. Then there exists a unique k -scattered random measure {M(A)}, A ∈ B0 , such that for each t 0 (34)

d

M((0, t]) = B0 (t).

Proof. It is the same as the proof for the case k=1 in Nguyen ([10], Theorem 4.2). Definition 3.3. Let M be a k -scattered random measure defined by the equation (33). Then for any 0 s < t the quantities M((s, t]) are called k -increments of the Bessel process {B0t }. By the same reasoning as in Nguyen ([10], Theorem 4.3) we have Theorem 3.3. Every k-dimensional Bessel process B0t , t 0 is a stationary independent k -increments process. Now we proceed to construct a new non-linear stochastic integration of a nonnegative function w.r.t. a Bessel process. For simplicity we assume that k=1 and write the Bessel process started at 0 as B(t), t . Let M denote the ◦-scattered random measure associated with B(.) and let L 2+ [0, T], T > 0

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the Hilbert space of all measurable nonnegative functions f on [0,T] such that T 2 f (u)2 du < ∞. (35) f := 0

Given a partition Π := {t0 = 0 < t1 < ... < tN T} of an interval [0, T], T > 0 we put fΠ (t) =

(36)

N

fti χ(ti ,ti+1 ](t) .

i=0

Then, the integral

T 0

fΠ (t)d◦ B(t) is defined as

1

(37) 0

The integral

T 0

d

fΠ (t)d◦ B(t) =

N

fti B([ti , ti+1 ))).

i=1

f (t)d◦ B(t) is defined as:

T

(38) 0

ξ(t)d◦ B(t) = lim|Π|→0

N

fi M(ti , t(i+1) ),

i=1

where |Π| := max{ti+1 − ti , i = 0, 1, ...N} and the limit is taken in the distribution sense, provided it exists. Theorem 3.4. For each function f ∈ L2+ [0, T] the integral (36) exists in the T convergence in distribution and for any α > 0 the rad.ch.f. of S := 0 α f (u)d◦ B(u) is given by

T

− log EΛs (vS) = v2

(39)

f 2 (u)du,

0

v 0. Proof. We have (40)

− log EΛs (v

N i=1

N fi M(ti , ti+1 ) = v (ti+1 − ti ) fi2 2

i=1

→v

T

2 0

f 2 (u)du

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which implies the conclusion of the theorem. By the above definition and by using the rad.ch.f. we get the following theorem: Theorem 3.5. (i) Let f 1 , f2 ∈ L2+ [0, T] and c 0. We have

T

(41) 0

cd◦ B(t) = cB(T);

(ii) If supp( f1 ) ∩ supp( f2 ) = ∅, then independent and

T

T 0

◦

T

{ f1 (t)(t) + f2 (t)}d B(t) =

(42)

f1 (t)d◦ B(t) and

◦

T

f1 (t)d B(t) +

0

0

T 0

f2 (t)d◦ B(t) are

f2 (t)d◦ B(t)

0

(iii) ( non-linearity) In general

T

◦

T

{ f1 (t)(t) + f2 (t)}d B(t)

(43) 0

iii If fn → f

◦

T

f1 (t)d B(t) + 0

f2 (t)d◦ B(t).

0

in L2+ [0, T], then

T

T

◦

fn (t)d B(t) →

(44) 0

f (t)d◦ B(t)

0

in distribution. References 1. Bingham, N. H., Random walks on spheres, Z. Wahrscheinlichkeitstheorie Verw. Geb., 22 (1973), 169–172. 2. Cox, J. C., Ingersoll, J. E. Jr., and Ross, S. A., A theory of the term structure of interest rates. Econometrica, 53(2), 1985. 3. Jeanblanc, M., Pitman, J., and Yor, M., Self-similar processes with independent increments associated with Lévy and Bessel processes, 100, No.1-2 (2002), 223– 231. 4. Kalenberg, O., Random measures, 3rd ed. New York: Academic Press 1983. 5. Kingman, J. F. C., Random walks with spherical symmetry, Acta Math., 109 (1963), 11–53. 6. Bebedev, N. N., Special functions and their applications. Prentice-Hall, INC Englewood Cliffs, N.J. 1965. 7. Levitan, B. M., Generalized translation operators and some of their applications, Israel program for Scientific Translations, Jerusalem 1962. 8. Nguyen, V. T., Generalized independent increments processes, Nagoya Math. J. 133 (1994) 155–175.

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9. Nguyen, V. T., Generalized translation operators and Markov processes, Demonstratio Mathematica, 34 No 2, 295–304. 10. Nguyen, V. T., A convolution approach to Bessel processes. submitted to Urbanik Volume Prob. Math. Stat. 2006. 11. Raiput, B. S., Rosinski, J., Spectral representation of infinitely divisible processes, Probab. Th. Rel. Fields 82(1989), 451–487. 12. Revuz, D. and Yor, M., Continuous martingals and Brownian motion. Springerverlag Berlin Heidelberg 1991. 13. Sato, K., Lévy processes and infinitely divisible distributions, Cambridge University of Press 1999. 14. Shiga, T., Wantanabe, S., Bessel diffusions as a one-parameter family of diffusion processes, Z. Warscheinlichkeitstheorie Verw. geb. 27 (1973), 34–46. 15. Urbanik, K., Generalized convolutions, Studia math., 23 (1964), 217–245. 16. Urbanik, K., Cramér property of generalized convolutions, Bull. Polish Acad. Sci. Math. 37 No 16 (1989), 213–218.

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Spectral Representation of Multiply Selfdecomposable Stochastic Processes and Applications∗ Dedicated to Professor Dang Dinh Ang for his 80th birthday

Nguyen Van Thu1 , To Anh Dung2 , Duong Ton Dam2 , and Nguyen Huu Thai3 1

Department of Mathematics, International University, HCM City, email:[email protected] 2 Department of Mathematics, University of natural sciences, HCM City 3 Department of Mathematics and Statistics, University of Economics, HCM City

In the present paper we study multiply selfdecomposable probability measures (SDPM) and processes and prove their integral representations. Similarly, the multiple s-selfdecomposability case is treated. Our results extend some of known results due to Urbanik, K., Jurek, Z., Rosinski, J. and Rajput, B. S. As an application, following Cartea and Howinson ([1]) we introduce the DampedLévy-mixed - stable process which leads to a mathematical model for option pricing. AMS 2000 subject classification: Primary 60E07, 60B12,60G10; Secondary 60G51,60 H05,60E07. Key words: infinitely divisible processes, α-SDP, random measures, stochastic integration, α-s-SDPM

1. Introduction, Notation and Preliminaries The main aim of this paper is to prove that each multiply selfdecomposable process (MSDP) on an Euclidean space admits a stochastic integral w.r.t.a MSD random measure (RM). Moreover, we will consider similar problems for multiply s-self-decomposable processes (MsSDP). ∗ The paper is completed during the first and second author’s stay at the Department of Mathematics Ritsumeikan University.

245

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Through the paper we shall denote by X a fixed d-dimensional (d=1,2,...) Euclidean space with the usual inner product and norm .. Let P(X) denote the class of all probability measures (PM) on the σ-field B(X) of Borel subsets of X equipped with the weak convergence. Let us denote by ID(X) the class of all IDPM’s on the space. Given a positive number c we define on X the following two families of mappings Tc and Ur as follows: ⎧ ⎪ ⎪ ⎨Tr x = rx, (1) ⎪ x ⎪ ⎩Ur = max(0, x − r) x , Ur (0) = 0. Further, for a PM µ ∈ P(X) and a mapping T on X let Tµ denote the image of µ under T. Recall (cf. Loéve [12] and Sato [25]) that a PM µ ∈ P(X is called SD if for each 0 < c < 1 there exists a PM µ c such that (2)

µ = T c µ ∗ µc

where ∗ denotes the ordinary convolution of PM’s. The concept of shrinking SDPM (shortly, s-SDPM) was introduced by Medgyessy [14] and studied by Jurek [3], [4], [6]. Namely, a PM µ is called s-SD if it is ID and for each 0 < c < 1 there exists a PM µ c such that (3)

µ = U c µc ∗ µc

where the power is taken in the convolution sense. It is known [3], [4], [6], [29], [17] that if µ is SD (resp., s-SD) then µ, µ c are both ID. The class of all SDPM’s (resp., s-SDPM’s) on X is denoted by L(X) (resp., U(X)). Let Ln (X), n = 1, 2, ...(resp., Un (X), n = 1, 2, ...) denote the class of all n-times SDPM’s (resp., n-times s-SDPM’s) which were first introduced by Urbanik1 [29] (resp., Jurek [4] and then studied further by many other authors (cf., for example [4], [17], [25]...). They are defined recursively as follows: A p.m. µ ∈ Ln (X), n = 2, 3, ... if and only if µ ∈ L1 (X) and for each c ∈ (0, 1) the component µ c in (2) belongs to Ln−1 (X). µ

It has been proved by Nguyen ([18], Proposition 1.1) that a p.m. belongs to Ln (X), n = 1, 2, ..., if and only if, for every c ∈ (0, 1) there

1 It should be noted, that our notation L (X) used here and in references [17], [18] is other n than that in Urbanik and other Authors [4], [25]. In particular, in our notation, L1 (X) denotes the set of all SDPM’s on X while in [4], [25] this class was denoted by L0 (X).

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exists a p.m. ν := µc,n ∈ ID(X) such that the following equality holds: ∗rk,n µ = ∗∞ k=0 (Tck ν)

(4)

where the power is taken in the convolution sense and, for n = 1, 2, ... ; k = 0, 1, 2,... we put n+k−1 . (5) rk,n = k The formulas (4) and (5) lead to the following interpolation of classes L n (X) (cf. Nguyen [18] and [19]): For each α > 0 we put ⎧ ⎪ ⎪ α ⎨1 k = 0, (6) =⎪ ⎪ ⎩α(α − 1)...(α − k + 1)/k! k = 1, 2, ... k and introduce the class α-times SDPM’s, shortly, α − SDPM s as the following: Definition 1.1. (cf. Nguyen [19]) A p.m. µ ∈ L α (X), α > 0, i.e. it is α-times SDPM’s, if and only if, for every c ∈ (0, 1) there exists a p.m. ν := µ c,α ∈ ID(X) such that the following equality holds: rk,α µ = ∗∞ k=0 (Tck ν)

(7)

where the power is taken in the convolution sense and, for any α > 0 and k = 0, 1, 2,... we put α+k−1 (8) rk,α = k It should be noted (cf. [19]) that the infinite convolution (7) is weakly convergent if and only if (9) logα (1 + x)ν(dx) < ∞ X

In the sequel, we shall denote by IDlogα (X) the subclass of ID(X) of all distributions for which the condition (9) is satisfied. Further, If {X(t)} is a X-valued Lévy process d

with ν = X(1) and ν ∈ IDlogα (X), then we say that it is of the class ID logα (X). Now, let us quote the following important integral representation for SDPM’s due to Vervaat-Jurek [5]: Theorem 1.1. (Jurek-Vervaat) A p.m. µ belongs to L 1 (X) if and only if there exists an X-valued Lévy process {X(.)} of the class ID log such that ∞ d (10) µ= exp(−t)X(dt) 0

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The integrator {X(.)} is called the background driving Lévy process (shortly,BDLP) of µ (cf. [5], [6]) and the r.v. X(1) is called BD r.v.. Further, Nguyen, N. H. [16], obtained the following pretty generalization of Theorem 1.1 to the case of α−SDPM’s for each α > 0. Theorem 1.2. (Nguyen, N. H. [16]) A p.m. µ belongs to L α (X) if and only if there exists an X-valued Lévy process {Xα (t)} of the class IDlogα (X) such that ∞ d (11) µ= exp(−t)tα−1Xα (dt) 0

In the sequel we shall need the following representation of ch.f.’s of ID and MSDPM’s on X: Theorem 1.3. (cf [20], [24]) A p.m. µ is ID if and only if its ch.f. µ(y), ˆ y ∈ X is of the unique form: (ei − 1 − iτ(x))M(dx) (12) − log µ(y) ˆ = i < z, y > + < Σy, y > − X

where z ∈ X is fixed; Σ is a quadratic form on X and M is a Lévy measure on X characterized by the property that M(0) = 0 , M is finite outside of very neighborhood of the origin and x2 M(dx) < ∞; 2 U1 1 + x the function τ(x) is defined by

⎧ ⎪ ⎪ ⎨x x ∈ U1 ; τ(x) = ⎪ ⎪ ⎩1 x > 1,

U1 being the closed unit ball in X. In what follows, if µ is ID with the ch.f. given by (12) then we will identify it with the triple [z, Σ, M]. Thus,we have Theorem 1.4. (cf. Nguyen [19], Theorem 2.4) A p.m. µ belongs to L α (X), α > 0 if and only if µ = [z, Σ, M], where z, Σ are the same as in Theorem (1.3) and the Lévy measure M is given by ∞ vα (x)( χA (e−u x)uα−1 du)m(dx) (13) M(A) = X

0

where m is a finite measure on X vanishing at the origin; A is a Borel subset of the real line separated from 0; the weight function vα (x) is defined by ∞ e−2t x2 α−1 −1 t dt (14) vα (x) = 1 + e−2t x2 0

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Theorem 1.5. (Nguyen [17], [19]) (A p.m. µ is mixed-stable i.e. µ ∈ L ∞ (X) if and only if µ = [z, Σ, M], where z, Σ are the same as in Theorem (1.3) and the Lévy measure M of µ is given by ∞ dt χA (tx) 2x+1 h(x)ν(dx) (15) M(A) = t V1 0 where ν is a PM on the open unit ball V1 := {x ∈ X : x < 1} and h(x) is a nonnegative continuous weight function on V1 . (α)

2. Mappings {T c } and Classes {Lα (X)} (α) . In this section we introduce families of mappings {Tc },where 0 < c < 1; α > 0, acting on the whole class ID(X) and show that they play the same role as mappings Tc in the definition of α-SDPM’s. To begin with let us consider the following particular cases: 2.1 α = n = 1, 2, ... Let µ ∈ Ln (X), n = 1, 2, .... By Proposition 1.1 [19], for every 0 < c < 1 Eq. (4) holds. Putting rk,n Tcn µ = ∗∞ k=1 (Tck ν)

(16)

and taking into account (4) we have (n)

µ = Tc µ ∗ µc,n

(17)

Conversely, it is also true. Namely, by induction one can prove that if a PM µ satisfies Eq. (17) for each 0 < c < 1 and for a PM µ c,n , then it belongs to Ln (X). 2.2 0 < α < 1. This case was treated in [19]. Namely, for such α the mapping T c,α is defined in [19]. Then, by Theorem 2.1 [19], it follows that a PM µ belongs to L α (X) if and only if for every 0 < c < 1 there exists a PM µc,α such that µ = Tcα µ ∗ µc,α

(18) 2.3 The general case α > 0 : It is easy to show that (19)

1=

∞ ∞ (−1)k−1 rk,α = |rk,α | k=1

k=1

Consequently, the mapping Tc,α : ID(X) → ID(X) given by (20)

|rk,α | Tc,α µ = ∗∞ k=1 Tck µ

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for any 0 < c < 1 and α > 0 is well-defined. Furthermore, the following general theorem holds: Theorem 2.1. A PM µ belongs to Lα (X), α > 0, if and only if for each 0 < c < 1 there exists a PM µc,α such that Eq. (18) holds. Proof. The ”if” part is similar to the proof of Theorem 2.1 [19]. To prove the ”only if” part one may assume that α = β + n, where 0 < β < 1, n = 1, 2, .... But it is clear by virtue of the cases 2.1 and 2.2 and by noticing that the mappings Tc,n and Tc,β commute with each other. Theorem 2.2. (α-differentiability of α-SDPM’s on X) For every α > 0 and every PM µ ∈ Lα (X) there exists a weak limit, denoted by Dα µ, which belongs to IDlogα (X) and satisfies the equation −α

Dα µ = limt→0 µtc,α

(21)

where t = − log c, µc,α is as in (7) and (17). Proof. (See Nguyen [19], Theorem 2.4 ). Definition 2.1. (cf. Nguyen [19]) The limit measure D α µ in Theorem (2.2) is called the α-derivative of µ. The following Theorem is obvious: Theorem 2.3. For each α > 0 the operator D α stands for an algebraic isomorphism between Lα (X) and IDlogα (X). (α)

3. Mappings {U c } and Classes {U α (X)} Following verbatim the proof of cases 2.1, 2.2 and 2.3 we have the Theorem: Theorem 3.1. For any 0 < c < 1 and α > 0 and for every PM µ ∈ ID(X) we put (22)

α

α

|( k )|c |( k )| = ∗∞ Uc,α µ = ∗∞ k=1 Tck µ k=1 Uck µ k

Then we get a mapping U c,α which stands for a well defined continuous isomorphism of the convolution algebra ID(X). Moreover, restricted to ID(X), it stands for an analogue of the shrinking mapping U c in (1). Definition 3.1. A PM µ ∈ ID(X) is said to be of the class U α (X), α > 0, or equivalently, α-s-SD, if for each 0 < c < 1 the following formula holds: (23) for some PM µc,α ∈ ID(X)

µ = Uc,α µ ∗ µc,α

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From the above definition we have: Theorem 3.2. A PM µ = [z, Σ, M] belongs to Uα (X), α > 0 if and only if the Lévy measure M satisfies the following condition: (24)

Σ∞ k=0 |

α k |c Tck M ≥ 0 k

for each 0 < c < 1, or, equivalently, (25)

Σ∞ k=0 |

α |U k M ≥ 0 k c

Definition 3.2. (cf. Jurek [8]) Given α > 0 let G α denote a Gamma r.v. with distribution τα . Let U (X) denote the class of all distributions of tdYρ (τα (t)), where Yρ (.) is a X-valued Lévy process with L(Yρ (1)) = ρ. (0,1) By virtue of formulas (23) and (24) and ((29) in Jurek [8]) we have the following theorem Theorem 3.3. The following equation hold: (26)

Uα (X) = U

which shows that definitions 3.1 and 3.2 are equivalent. 4. Stochastic Representation of MSDPM’s and s-MSDPM’s. Our main aim in this Section is, following the method of Rajput and Rosinski[24], to give a representation of MSDC and s-MSDC processes via stochastic integrals w.r.t. the corresponding random measures (RM). Namely, since the general forms of the Lévy measures were obtained in Sections 2 and 3 the Kolmogorov extension theorem and the method of Rajput and Rosinski [24] allow to obtain the required representation. Definition 4.1. Let T be a parameter set Z of all integers or R of all real numbers. A stochastic process Xt , t ∈ T is said to be ID, stable, mixed-stable, α-SD, α-s-SD if for any t1 , t2 , ..., tn ∈ T and λ1 , λ2 , ..., λn , n = 1, 2, ... the r.v. Σn1 λ j Xt j is ID, stable, mixed-stable, α-SD, α-s-SD, respectively. Definition 4.2. Let Λ = {Λ(A) : A ∈ S} be a real stochastic process defined on a probability space (Ω, F , P), where S stands for a σ-ring of subsets of an arbitrary non-empty set S satisfying the following condition : There

exists an increasing sequence Sn , n = 1, 2, ... of sets in S with n Sn = S.

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We call Λ to be an independently scattered RM, if, for every sequence {An } of disjoint sets in S, the random variables Λ(An ), n = 1, 2, ... are independent, and , if ∪n An belongs to S, then we also have Λ(∪n An ) = Σn Λ(An ) a.s., where the series is assumed to be convergent a.s. In addition, if for every A ∈ S the distribution of Λ(A) is ID, stable, mixed-stable, MSD, respectively, then we say that it is an ID, stable, mixed-stable, MSD RM. Each r.v. Λ(A), A ∈ S has the ch.f. ∞ 1 2 (eitx − 1 − itτ(x))FA (dx). (27) − log E exp(itΛ(A) = itν 0 (A) + t ν1 (A) − 2 −∞ where t ∈ R, A ∈ S and − ∞ < v0 (A) < ∞, 0 ≤ v1 (A) < ∞ and FA is a Lévy measure on R. Moreover, v0 is a signed measure , v1 a measure and FA a Lévy measure. Moreover, we have the following Theorem 4.1. (cf. Raiput and Rosinski [24], Proposition 2.1) The ch.f. (27) can be written in the unique form: (28) E exp(itΛ(A)) = exp( K(t, s)λ(ds)) A

where t ∈ R, A ∈ S and (29)

2 2

(eitx − 1 − itτ(x))ρ(s, dx),

K(t, s) = ita(s) − 1/2t σ (s) + A

with (30)

a(s) =

dv0 (s) dλ

σ2 (s) =

dv1 (s) dλ

and (31)

and ρ is given by Lemma 2.3 in (cf. Raiput and Rosinski [24]. Moreover, we have (32) |a(s)| + min{1, x2}ρ(s, dx) = 1 a.e.λ. R

Definition 4.3. (cf. Urbanik and Woyczynski [27])

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(a) If f is a simple function on S, f = Σ j x j χA j , A j ∈ S then, for each A ∈ σ(S), we put f dΛ = Σ j λ(A ∩ A j )

(33) A

(b) A measurable function f : (S, σ(S)) → (R, B(R) is said to be Λ−integrable if there exists a sequence { fn } of simple functions as defined in (a) such that (i) fn → f

a.e. λ,

(ii) For every A ∈ σ(S), the sequence { n → ∞. If f

f dΛ} A n

converges in prob., as

is Λ−integrable, then we put

f dΛ = P − limn→∞

{ A

fn dΛ, A

where { fn } satisfies (i) and (ii). Now, combining Theorems 1.3, 1.4, 1.5 we get the following: Theorem 4.2. Given α > 0, let Λ(A), A ∈ S be a α − s.d.r.m. Then, the ch.f. of Λ(A) is of the unique form (20) where (34)

⎧ ⎪ ⎪ s) = ita(s) − 1/2t 2σ2 (s) ⎨K(t, ⎪ ⎪ ⎩+ (eitx − 1 − itτ(x))ρ(s, dx) A

with (35)

a(s) =

dv0 (s) dλ

σ2 (s) =

dv1 (s) dλ

and (36) Moreover, we have |a(s)| +

R

min{1, x2}ρ(s, dx) = 1 a.e.λ.

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Proof. By virtue of (13) it follows that for any A ∈ S and t ∈ R Λ(A) has the representation (37) 1 − log E exp(itΛ(A)) = itν 0 (A) + t2 ν1 (A) − 2

∞

−∞

vα (x)(

∞

k(e−u x, t)uα−1 du)m(A, dx)

0

which, by a similar argument of Proposition 2.1 in Rajput and Rosinski [24], implies that there exists a unique finite measure ν on σ(S) × B(R) such that ν(A × B) = m(A, B),

for any A ∈ S, B ∈ B(R).

Moreover, for every A ∈ σ(S) we have ν(A, {0}) = 0. Now, we are in the position to present the following theorem whose proof is a simple combination of Theorem 6 and the Komogorov extension theorem and Theorem 5.2 in Rajput and Rosinski [24]. Theorem 4.3. Given 0 < α ≤ ∞ let {Xt : t ∈ T} be an α − SD stochastic process defined on a probability space (Ω , P ). Then there exists an α − SDRM, say Λ, defined on the probability space (Ω, P) such that Ω = Ω × I, P = P × Leb , Leb being the Lebesgue measure on I and {Xt : t ∈ T} = { ft (s)dΛ(s) : t ∈ T} a.s.P, S

where { ft (s) : t ∈ T, s ∈ S} are some measurable functions on S and I denotes the closed unit interval. By a similar argument as for MSDPM’s we have the following: Theorem 4.4. Given α > 0 let {Xt : t ∈ T} be an α − s − SD stochastic process defined on a probability space (Ω , P ). Then there exists an α − s − SDRM, say Λ, defined on the probability space (Ω, P) such that Ω = Ω × I, P = P × Leb , Leb being the Lebesgue measure on I and {Xt : t ∈ T} = { ft (s)dΛ(s) : t ∈ T} a.s.P, S

where { ft (s) : t ∈ T, s ∈ S} are some measurable functions on S and I denotes the closed unit interval.

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5. An Application in Option Pricing If X is Lévy - stable random variable with index 0 < α < 1, then it does not have any integer moments, and, for the case 1 < α < 2, only the first integer moment exists. Therefore, to overcome this difficulties, following Cartea and Howinson [1], we introduce the following Damped-Lévy-mixed - stable process which will lead to a mathematical model for our purpose of option pricing. Suppose that X j (t), j = 1, 2 are independent Lévy -stable processes with indexes 0 < α1 < α2 < 2, respectively such that the logarithm of the characteristic function of X j (1) is given by +∞ (eiux − 1 − iuτα j (x))W j (x)dx, j = 1, 2. (38) ψ j (u) = −∞

where

and

⎧ ⎪ ⎪ ⎨Cq |x|−1−α j W j (x) = ⎪ ⎪ ⎩Cp x−1−α j ⎧ ⎪ x ⎪ ⎪ ⎪ ⎨ τα j (x) = ⎪ sinx ⎪ ⎪ ⎪ ⎩0

for x < 0 for x > 0 for α j > 1 for α j = 1 for α j < 1.

Here Cp , Cq > 0 are scale constants, p, q ≥ 0 and p + q = 1. Following Cartea and Howinson[1] the exponential cut-off e −λ|x| is introduced to obtain the Damped Lévy measures ⎧ ⎪ ⎪ for x < 0 ⎨Cq |x|−1−α j e−λ|x| , (39) W λj (x) = ⎪ . ⎪ ⎩Cp x−1−α e−λ|x| , for x > 0 Let W λj , j = 1, 2, denote the Damped Lévy measures corresponding to Lévy processes Xλj (t), j = 1, 2 with (40)

φλj (u)

=

+∞ −∞

(eiux − 1 − iuτα j (x))e−λ|x| W j (dx)

Putting, for t ≥ 0, X(t) = X 1 (t) + X2 (t) we get a Lévy process X(t) which is also a mixed-stable-Lévy process with Φ(u) = Φ 1 (u) + Φ2 (u), where Φ j (u), j = 1, 2 are given by (40). Putting ⎧ ⎪ ⎪ for x < 0 ⎨Cq |x|−1−α j e−λ|x| , (41) W λj (x) = ⎪ , j = 1, 2 ⎪ ⎩Cp x−1−α j e−λ|x| , for x > 0 and taking into account (40) we infer that the logarithm of the ch.f., denoted by φλ (u), for a Damped-Lévy process {X λ (t)} is of the form (42)

φλ (u) = φλ1 (u) + φλ2 (u)

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where (43)

φλj (u)

=

+∞ −∞

(eiux − 1 − iuτα j (x))e−λ|x| W λj (dx), j = 1, 2.

The Damped-Lévy process X λ (t) := X1λ (t) + X2λ (t) has the following property: (i) {Xλ (t)} is a Lévy process. (ii) It is not a stable process. (iii) limλ→0 Xλ (t) = X(t) (in distribution and in probability). (iv) The process {Xλ (t)} has finite moments of all orders. Moreover, its exponential moments exist. Suppose that we work under the framework of the market with the stock price process X(t) = X 1 (t) + X2(t) which satisfies the condition that X j (t), j = 1, 2 are independent α j -stable Lévy processes under measure Q. Our further aim is to deduce a kind of the Black-Scholes formula under Lévy-Mixed-Stable Shocks. In what follow we assume that 1 < α1 < α2 < 2. Then, by Cartea and Howinson ([1], Proposition 3, p. 12) we have (44)

φλj (θ) = κα j {p(λ − iθ)α j + q(λ + iθ)α j − λα j − iα j αα j −1 (q − p)θ},

(j=1,2). We assume that the logarithm of the stock price process, under the risk-neutral measure, is a Damped-mixed-stable Lévy process. Then, by Cartea and Howinson ([1]) (45)

St+∆t = St exp(r−D0 )∆t−φ(−iσ)+σφ ,

where r > 0 is the risk free rate and σ > 0. Equation (45) can be rewritten as the following: (46)

St+∆t = St expπ∆t+σφ

where π, φ are parameters for Damped-(α1, α2 )-mixed-stable-Lévy process X λ (t). Then as ∆t → 0 the ”Damped Black-Scholes” PDE can be given and solved as in the Damped-stable-Lévy case (cf. Cartea and Howinson ([1]), p. 24).

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References

´ and Howinson, S., Distinguished limits of Lévy—stable process, and ap1. Cartea, A. plications to option pricing, Oxford Financial Research Centre, Ser. OFRC working papers Ser., No. 2002mf04. 2. Hardin, Jr., C. D., On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12, 385–401, 1982. 3. Jurek, Z. J., Limit distributions for sums of shrunken random variables, Dissertationes Mathematicae, 185, PWN Warszawa, 1981. 4. Jurek, Z. J., Limit distributions and one-parameter groups of linear operators on Banach spaces, J. Multivariate Anal. 13, 578–601, 1983. 5. Jurek, Z. J. and Vervaat, W., An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrsch. verw. Gebiete. 62, 247–262, 1983. 6. Jurek, Z. J., Selfdecomposability perpetuity laws and stoping times, Probab. Math. Stat. 19, 413–419, 1999. 7. Iksanov. M. A., Jurek, Z. J., and Schreiber, M. B., A new factorization property of the selfdecomposable probability measures, Annals of Probability, 32, No. 2, 1356–1369, 2004. 8. Jurek, Z. J., The random integral representation hypothesis revisited: new classes of s-selfdecomposable laws. Proc. Int. Conf. Hanoi, 13–17 August 2002, World Scientific, Hongkong 2004, 495–514. 9. Kallenberg, O., Random measures, 3rd ed. New York: Academic Press 1983. 10. Kuelbs, J., A representation theorem for symmetric stable processes and stable measures on H. Z. Wahrscheinlichkeits, Verw. Geb. 26, 259–271, 1973. 11. Kumar, A. and Schreiber, B. M., Characterization of Subclasses of Class I, Probability Distributions, Ann. Probab. 6, 279–293, 1978. 12. Loéve, M., Probability theory, New York, 1950. 13. Maruyama, G., Infinitely divisible processes, Theor. Prob. Appl. 15, 3–23, 1970. 14. Medgyessy, P., On a new class of unimodal infinitely divisible distributions and related topics. Studia Sci. Math. Hungar. 2, 441–446, 1967. 15. Musielak., Orlicz spaces and modular spaces, Lecture Notes math., vol. 1034, New York Berlin Heidelberg: Springer 1983. 16. Nguyen, N. H., Stochastic representation of α-times selfdecomposable diistributions., a private communication. 17. Nguyen, V. T., Stable type and completely self-decomposable probability measures on Banach spaces, Bull, Ac. Pol., Sér. Sci. math., 29, No. 11–12, 1981. 18. Nguyen, V. T., Fractional Calculus in Probability, Probab. Math. Statst. 3 No. 2, 171–189, 1984. 19. Nguyen, V. T., An alternative approach to multiply self-decomposable probability measures on Banach spaces, Probab. Th. Rel. Fields 72, 35–54 (1986). 20. Parasarathy, K. R., Probability measures on metric spaces, New York: Academic Press 1967. 21. Prékopa, A., On stochastic set functions I, Acta Math. Acad. Sci. Hung. 7, 215–262, 1956. 22. Prékopa, A., On stochastic set functions II, III. Acta Math. Acad. Sci. Hung. 8, 337–400, 1957. 23. Rosinski, ´ J., Random integrals of Banach space valued functions, Studia math. 78,

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15–38, 1985. 24. Rajput, B. S. and Rosinski, ´ J., Spectral representation of infinitely divisible processes, Probab. Th. Rel. Fields 82, 451–487 (1989). 25. Sato, K., I., Urbanik’s, Class L m of Probability Measures, Ann. Sci. Coll. Lib. Arts Kanazawa Uni. 15, 1–10, 1978. 26. Sato, K. I., Lévy processes and infinitely divisible distributions, Cambridge University of Press 1999. 27. Urbanik, K. and Woyczynski, W. A., Random integrals and Orlicz spaces, Bull. Acad. Polon. Sci. 15, 161–169(1967). 28. Urbanik, K., Random measures and harmonizable sequences, Studia math. 31, 61–88, 1968. 29. Urbanik, K., Slowly varying sequences of random variables, Bull. Acad. Pol. Sci. Série des Sci. Math. Astr. et Phys. 20, 8(1972), 679–682.

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EconoGrowth

Stochastic Growth Models of an Isolated Economy Kunio Nishioka

Key words: Neo-classical model, Economic growth with uncertainty, the stochastic Solow equation, a diffusion process on (0, ∞) 1. Introduction Study of economic growth has continued more or less steadily for almost 600 years after A. Smith and T. Malthus. The main thesis of economic growth theory is to answer the following question: Why some nations are so rich and the others are so poor? In nowadays economic growth theories, Neo-classical growth model plays the fundamental part. This model was developed by the works of R. Solow, 1956, 1957. After Solow’s work, Lucus (1988), Romer (1986), Mankiw (1992), and etc. refined Solow model by importing advances of technology or human factor. Today a mainstream is an endogenous growth theory, which build these advances into economic growth itself. We will start from Solow model. An economy in the model is considered in the following setting. Assumption 1.1. (i) The economy is an isolated island in where many labors live. There is a social planner, who governs all economic. (ii) There is one good. At time t, production Y(t) of the good depends on two factor, capital K(t) and labor L(t). The good can be either consumed or invested as capital. (iii) The social planner saves a constant fraction s ∈ (0, 1) of production, to be added to the economy’s capital stock, and distributes the remaining fraction uniformly across the labors of the economy. In what follows, we introduce the following normal signatures in economic theory: 259

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260

(1.1)

Y(t) = output at time t, I(t) = investment at time t,

K(t) = capital stock at time t, C(t) = consumption at time t,

L(t) = the number of labors at time t. From Condition 1.1 ( + a little ), the following condition is derived: Condition 1.2. (i) The economy is Keynes system, that is I(t) + C(t) = Y(t). (ii) The technology for producing the good is given by the production function F : R2+ → R+ , that is Y(t) = F(K(t), L(t)).

(1.2)

(iii) Capital depreciates at a fixed rate λ ∈ [0, 1], that is K (t) = I(t) − λ K(t). (iv) Saving rate s ∈ (0, 1) is constant, that is Y(t) = s Y(t) + C(t). (v) The population of labors increases in a constant rate n: L (t) = n L(t).

(1.3)

In addition, we assume that the production function F in (1.2) is neo classical, i.e. the following condition is fulfilled. Condition 1.3. The production function F is a strictly concave C 2 class function with F(0, L) = 0 = F(K, 0). Moreover F satisfies: (i) Inada Condition: lim ∂K F(K, L) = ∞,

K→0

lim ∂K F(K, L) = 0,

K→∞

lim ∂L F(K, L) = ∞, L→0

lim ∂L F(K, L) = 0.

L→∞

(ii) CRS condition (constant returns to scale): F(a K, a L) = a F(K, L) for ∀a > 0.

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Example 1.4. A typical example of the above production function is F(K, L) = Kα L1−α ,

0 < α < 1,

which is called Cobb-Douglus type. We introduce the per-capita measurements, that is (1.4)

y(t) ≡ Y(t)/L(t) k(t) ≡ K(t)/L(t)

(per-capita GDP), (per-capita capital stock),

By CRS condition in Condition 1.3, (ii), (1.5)

y(t) =

Y(t) F(K(t), L(t)) K(t) = = F( , 1) ≡ f (k(t)) L(t) L(t) L(t)

We also call this f as a production function. By definition (1.5) of f and Condition 1.3, Condition 1.5. A production function f : R + → R+ is a strictly concave C2 class function with f (0) = 0. Moreover f satisfies (1.6)

(Inada condition)

lim f (k) = ∞, k→0

lim f (k) = 0.

k→∞

Combining the equations in Condition 1.2, we derive ODE for the capital stock K(t): K (t) = Y(t) − C(t) − λ K(t) = s Y(t) − λ K(t) (1.7) = s F K(t), L(t) − λ K(t). By a simple calculation, k (t) =

K(t) L(t)

=

K (t) K(t) L (t) − · . L(t) L(t) L(t)

Now (1.7) and (1.3) give the dynamics of capital stock in per-capita measurement: (1.8) (Solow equation) k (t) = s f (k(t)) − λ + n k(t), where s ∈ (0, 1) is saving rate, λ ∈ [0, 1] is capital depreciating rate, and n is population growth rate.

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Owing to Condition 1.5, there exists a unique solution k∗ to (1.9)

s f (k) − (λ + n) k = 0,

k > 0,

and it is a stable fixed point of Solow equation (1.8). Proposition 1.6. There exists a unique point k∗ > 0 which solves (1.9). We call k∗ as the state of golden age, since lim k(t) = k∗

t→∞

for any k(0) > 0.

2. Verification of Solow model We shall compare the result in Proposition 1.6 with a statics in the real economy between 1980 and 1997. Growth rate of per-capita GDP is y (t)/y(t) which is easily derived from (1.5) and Solow equation (1.8): (2.1)

f (k(t)) y (t) k(t) = k (t) = f (k(t)) s − (λ + n) y(t) f (k(t)) f (k(t))

From Condition 1.5, k/ f (k) ∼ 1/ f (k) ∼ 0 if k is sufficiently small, and we know that the right hand side of (2.1) behaves as k f (k) s − (λ + n) f (k)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

diverges if k → 0, > 0 and monotonely decreases in k if 0 < k < k∗ , =0 if k = k∗ , k∗ .

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Figure 2.1. Per-capita GDP and its growth rate

From the above, the following observations are easily derived: (i) If k(t) is small, y (t)/y(t) should be very large, (ii) If k(t) is near to the golden age k∗ , y (t)/y(t) should be very small. Therefore all dots in Fig. 2.1 should be distributed along the bold curve in the figure. But there exist many such counter examples in the zone A of Fig. 2.1 and it is difficult to justify nations with negative growth rate by Solow model.

3. The stochastic Solow equation I. In order to prevail the previous contrariety, many economists make various attempts to approve Solow model. (i) Lucus (1988), Romer (1986), and etc. imported advances in technology into Solow model. For instant, Lucus considered a production function (3.1)

Y(t) = F(K(t), A(t) L(t)),

where (3.2)

A(t) ≡ exp{gt},

g is a non-negative constant

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is an advances in technology for each labor. (ii) Uzawa (1965), Mankiw(1992), and etc. introduced human factor. For instant, Mankiw introduced a production function Y(t) = F(K(t), L(t), H(t)) with a human factor H(t), and he derived a simultaneous equations k (t) = sk y(t) − (n − λk )k(t), h (t) = sh y(t) − (n − λh )k(t), where sk is a constant saving rate to capital stock. sh is a constant saving rate to human capital stock, and λk , λh are constant depreciating rates. (iii) Some economists tried to randomize Solow equation. II. Especially in (iii), Merton (1975) shifted the population growth equation (1.3) on to a SDE1 (3.3)

dL(t, w) = n L(t, w) dt + σ L(t, w) dB(t, w),

where n and σ are positive constants and {B(t, w)} is a one dimensional Brownian motion. By Ito’s ˆ formula, Merton has obtained the following SDE which accounts per-capita capital stock {k(t, w)} as a diffusion process in (0, ∞):

(3.4)

(the stochastic Solow equation) dk(t, w) = −σ k(t, w) dB(t, w) + s f (k(t, w)) − λ + n − σ2 k(t, w) dt

We are interesting to precise behaviors of {k(t, w)} and its growth rate. The growth rate is defined as (3.5) k (t)/k(t) = log k(t) , when k(t) is a solution of (1.8). But k (t) has no sense in SDE, and we should consider an average growth rate in time2 (3.6) 1 Cho

ρ(t, w) ≡

log k(t, w) − log k(0) t

and Cooley (2001) replaced (3.2) by the diffusion process A(t, w) ≡ exp{(g − a2 /2)t + a B(t, w)}. 2 This converges to the Lyapunov index of {k(t, w)} as t → ∞.

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instead (3.5). Proposition 3.1. Let the production function f satisfy Assumption 1.6, and define a constant θ by θ ≡ λ + n − σ2 /2.

(3.7)

Then asymptotic behaviors of k(t, w) and its average growth rate ρ(t, w) are as follows: θ0 k(t, w) → ∞ a.s. recurrent† recurrent ρ(t, w) → − θ a.s. → 0 a.s. Here ‘ recurrent’ means that {k(t, w)} is a recurrent diffusion on (0, ∞) with an invariant probability measure. In ‘ recurrent†’ case, {k(t, w)} is recurrent but its invariant measure is infinite and it converges in Césaro’s sense, that is 1 lim T→∞ T

(3.8)

T

k(t, w) dt = ∞ a.s.

0

Remark 3.2. (i) Let θ ≥ 0. Then for the stochastic Solow equation case (3.4), there is no such state of golden age as in Proposition 1.6. Since k(t, w) is recurrent, k(t, w) reaches every point on (0, ∞) with probability one. (ii) Let K(t, w) be the total capital stock. Then it holds that log K(t, w) − log K(0) σ2 = lim ρ(t, w) + n − . t→∞ t→∞ t 2 lim

From Proposition 3.1 and the above remark, we have the main theorem. Theorem 3.3. Suppose that the production function f satisfies Assumption 1.6. On the (λ, n − σ 2 /2) plain, we define domains A through C as follows:

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Then asymptotic behaviors of the indicators, the per-capita capital stock k(t, w), the population of labors L(t, w), and

(3.9)

the total capital stock K(t, w), are as follows: ∈A ∈B ∈C (λ, n − σ2 /2) k(t, w) → ∞ a.s. recurrent recurrent L(t, w) → 0 a.s. → 0 a.s. → ∞ a.s. → 0 a.s. → 0 a.s. → ∞ a.s. K(t, w)

4. A quaere to Inada condition I. In most neo-classical growth models, it is supposed that the production function satisfies Inada condition in Assumption 1.3. However some economists newly assert that Inada condition at zero lim f (k) = ∞

(4.1)

k→0

is inapposite. From CRS condition and the mean value theorem, K+1 K , 1) − F( , 1) F(K + 1, L) − F(K, L) = L F( L L K+1 1 K = L f( ) − f ( ) = L f (y) = f (y), L L L When L is sufficiently large, y is small, and (1.6) derives that f (y) ∞. This means that under the assumption (4.1), an additional unit of capital derives any large production if an amount of labour is sufficiently large, what conflicts to the real economic data. So Kamihigashi (2003) proposed to suppose the following (4.2) instead of (4.1): Condition 4.1. The production function f satisfies (4.2)

C2 class, strictly concave, and f (0) = 0, 0 < ∃ lim f (k) ≡ f (0) < ∞, lim f (k) = 0. k→0

k→∞

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Example 4.2. We present a production function satisfying Condition 4.1, that is also a modification of Cobb-Douglus type: let c > 0 and 0 < α < 1 be constants and α F(K, L) ≡ K + c L L1−α − cα L. From this F, we have f (k) ≡ (k + c)α − cα .

Proposition 4.3. Let the production function f satisfy Assumption 4.1. Set the constant θ as in (3.7) and a constant γ as γ ≡ s f (0) − θ = s f (0) − (λ + n −

σ2 ). 2

Then θ θ s f (0) k(t) recurrent‡ → 0 a.s. ρ(t) → 0 a.s. → γ a.s. Here ‘ recurrent’ means that {k(t, w)} is a recurrent diffusion on (0, ∞) with an invariant probability measure. In ‘ recurrent†’ and ‘ recurrent‡’ case, {k(t, w)} is recurrent with an infinite invariant measure and converges in Césaro’s sense, that is 1 lim T→∞ T

T

k(t) dt = 0

∞ a.s. 0 a.s.

recurrent† recurrent‡ .

II. We shall investigate asymptotic behaviors of economic indexes (3.9). Case 1: s f (0) ≥ 1. Theorem 4.4. Suppose that the production function f satisfies Assumption 4.1 and that s f (0) ≥ 1. On the (λ, n − σ 2 /2) plain, we define domains A through D as follows:

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Then asymptotic behaviors of economic indexes (3.9) are as follows: (λ, n − σ2 /2) ∈A ∈B ∈ C k(t, w) → ∞ a.s. recurrent recurrent L(t, w) → 0 a.s. → 0 a.s. → ∞ a.s. → 0 a.s. → 0 a.s. → ∞ a.s. K(t, w) (λ, n − σ2 /2) ∈ D k(t, w) → 0 a.s. L(t, w) → ∞ a.s. K(t, w) → ∞ a.s.

Case 2: 0 < s f (0) < 1. Theorem 4.5. Suppose that the production function f satisfies Assumption 4.1 and that 0 < s f (0) < 1. On the (λ, n − σ2 /2) plain, we define domains A through F as follows:

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Then asymptotic behaviors of economic indexes (3.9) are as follows: ∈A ∈ B ∈C (λ, n − σ2 /2) k(t, w) → ∞ a.s. recurrent recurrent → 0 a.s. → 0 a.s. → ∞ a.s. L(t, w) K(t, w) → 0 a.s. → 0 a.s. → ∞ a.s. (λ, n − σ2 /2) ∈D ∈ E ∈ F k(t, w) → 0 a.s. → 0 a.s. → 0 a.s. L(t, w) → ∞ a.s. → ∞ a.s. → 0 a.s. → ∞ a.s. → 0 a.s. → 0 a.s. K(t, w)

Appendix. One dimensional diffusion process with boundaries I. We shall review behaviors of the diffusion process {k(t, w)} defined by SDE (3.4). Fix an arbitrary point k0 ∈ (0, ∞), and define k (the scale function) S(k) ≡ ϕ(y) dy, k0

(density of the speed measure) m(k) ≡ where

y

ϕ(y) ≡ exp{−2 k0

σ2 k 2

s f (ξ) − (λ + n − σ2 ) ξ dξ}, σ2 ξ2

1 , · ϕ(k) y > 0.

Using the scale function S and the speed measure m(k) dk, Feller (1954) and Ito-McKean ˆ (1965) classified boundaries of a one dimensional diffusion into five types, that is a regular boundary, an entrance, an exit, an infinite natural, and a finite natural. II. For {k(t, w)} given by SDE (3.4), its boundary points are 0 and ∞, and both are natural boundaries. In this case, asymptotic behaviors is already known, Nishioka (1976). Case 1. Both are infinite natural: {k(t, w)} is recurrent on the interval (0, ∞), and density function of an invariant measure is m(k) 1 1 (A.1) µ(k) ≡ . = · 2 2 C C σ k ϕ(k)

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Here the constant C is ⎧ ∞ ⎪ ⎪ ⎪ m(k) dk if the integral is finite, ⎨ (A.2) C≡⎪ 0 ⎪ ⎪ ⎩ 1 otherwise. In addition, the following Ergodic Theorem holds: Maruyama-Tanaka (1957): If functions g, h are integrable with respect to µ(y) dy, then T ∞ g(k(t, w)) dt g(y) µ(y) dy 0 0 = ∞ a.s. (A.3) lim T T→∞ h(y) µ(y) dy h(k(t, w)) dt 0

0

where the denominator in the right hand side must not vanish. Case 2. One is finite natural and the other is infinite natural: (i) {k(t, w)} cannot reach boundaries within a finite time, almost surely. (ii)

⎧ ⎪ 0 a.s. if 0 is finite natural ⎪ ⎪ ⎪ ⎪ and ∞ is infinite natural, ⎨ lim k(t, w) = ⎪ ⎪ ∞ a.s. if 0 is infinite natural ⎪ t→∞ ⎪ ⎪ ⎩ and ∞ is finite natural.

Case 3. Both are finite natural: The statement (i) in Case 2 is true, but S(∞) − S(x) , S(∞) − S(0) S(x) − S(0) Px [lim k(t, w) = ∞] = . t→∞ S(∞) − S(0)

Px [lim k(t, w) = 0] = t→∞

A. Sketch of proofs Proof of Proposition 3.1 Step 1. Owing to Appendix §A, I, boundaries 0 and ∞ for {k(t, w)} are classified as follows: θ 0. In this case, an invariant measure is a probability measure and the law of iterated logarithm implies lim

T→∞

1 B(T, w) = 0 a.s. T

Then from (A.1) and Ergodic Theorem (A.3), ∞ f (k) σ2 lim ρ(T, w) = s µ(k) dk − (λ + n − ) a.s. T→∞ k 2 0 We calculate the first term in the right hand side. Put β ≡ 2(λ + n)/σ2 − 2. By Inada condition (1.6),

f (k) 2s k f (ξ) C exp{ dξ} k σ2 k2+β σ2 k0 ξ2 0 s C σ2 ∞ 1 2s k f (ξ) = dk 2 1+β exp{ 2 dξ} 2 2s σ k0 ξ σ k 0 k ∞ f (ξ) 1 C 2s = · 1+β exp{ 2 dξ} k=0 2 k σ k0 ξ2 ∞ 2 σ C 1 2s k f (ξ) (1 + β) + dk 2 2+β exp{ 2 dξ} 2 σ k0 ξ2 σ k 0

the first term = s

=

∞

dk

σ2 C 1 σ2 (1 + β) = λ + n − . 2 C 2

Now we have proved that limT→∞ ρ(T) = 0. Step 3. We shall investigate behavior of ρ(t, w) when θ = 0. In this case, an invariant measure µ(k) dk is not finite, that is ∞ ∞ 1 dk = ∞ for large L. µ(k) dk ∼ 2k σ L L Moreover the function f (k)/k may not be integrable.

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Fix an arbitrary ε > 0, and define a function h as h(k) ≡ εk,

k ≥ 0.

Since f satisfies Inada condition (1.6), we can find a unique point k† > 0 such that f (k) = h(k), k > 0.

Another functions f˜ and h are defined as

f (k) 0 ≤ k < k† ˜ f (k) ≡ h(k) ≡ ε k k† ≤ k,

f (k) − εk 0 ≤ k < k† ˜ h(k) ≡ f (k) − h(k) = 0 k† ≤ k. Here h(k)/k is integrable with respect to µ(k) dk. We easily see that T T≥ I(0,L) (k(t, w)) dt, 0

∞ 0

I(0,L) (k) µ(k) dk < ∞,

for arbitrary L > 0. By Ergodic Theorem (A.3), 1 T h(k(t, w)) 0 ≤ lim sup dt k(t, w) T→∞ T 0 T ∞ h(k(t, w)) h(k) dt µ(k) dk k(t, w) k 0 0 = ∞ ≤ lim T . T→∞ I (k) µ(k) dk I(0,L) (k(t, w)) dt (0,L) 0

0

∞

Note that

µ(k) dk = ∞, and let L → ∞. Then we have

0

1 lim T→∞ T

T 0

h(k(t, w)) dt = 0 a.s. k(t, w)

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Remark that 1 T

T 0

h(k(t, w)) 1 dt = k(t, w) T

T

ε dt = ε 0

From this and the previous calculation, 1 T f˜(k(t, w)) dt lim T→∞ T 0 k(t, w) 1 T h(k(t, w)) 1 T h(k(t, w)) = lim dt + lim dt = ε. T→∞ T 0 T→∞ T 0 k(t, w) k(t, w) Since the definition of f˜ asserts that 0 ≤ f ≤ f˜, 1 T f (k(t, w)) 0 ≤ lim sup dt k(t, w) T→∞ T 0 1 T f˜(k(t, w)) ≤ lim dt = ε a.s. T→∞ T 0 k(t, w) Here ε > 0 is arbitrary. Let ε ↓ 0 and we have 1 T f (k(t)) dt = 0 a.s. lim T→∞ T 0 k(t) Note that our assumption is θ = 0. Now we have lim ρ(T) = 0 − θ = 0 a.s.

T→∞

by an analogous way as in Step 1. We shall omit the remained proof.

2

Proof of Proposition 4.3 Using Appendix §A, I, we can classify the boundaries of {k(t, w)}: Put θ = λ + n − σ2 /2 and θ 0. Then, q q+2 2 z 2 q ˆ lim N min Z − Z 2 = Jq ( | f | q+2 dλq ) q , N

z

where PZ (dξ) = f (ξ)λq (dξ) + ν(dξ) is the Lebesgue decomposition of P Z with respect to the Lebesgue measure λq on Rq , and Jq is a constant depending on q, corresponding to the uniform distribution on [0, 1] q. Remark 4.1. In dimensions q = 1 and 2, J 1 = ∼

q 2πe

1 12

and J2 =

5√ . 18 3

For q ≥ 3, Jq

as q goes to infinity.

The optimal N-quantization problem that consists in determining a grid z , which minimizes the L2 -quantization error, relies on the property that the distorsion is continuously differentiable at any N-tuple having pairwise distinct components, with a gradient obtained by formal differentiation in (4.1) : (4.2) ∇DZN (z) = 2E KN (z, Z)], ∗

where KN : (Rq )N × Rq → (Rq )N is defined by KN (z, ξ) = ((zi − ξ)1ξ∈Ci(z) )1≤i≤N . A quantizer Zˆ = Zˆ z is said stationary if the associated N-tuple z satisfies ∇DZN (z) = 0. An optimal quantizer is a stationary quantizer. The integral representation (4.2) of ∇DZN suggests, as soon as independent copies of Z can be simulated, to implement a stochastic gradient algorithm (descent), in order to get numerically a stationary quantizer. By denoting, z(s) = (zs,1 , . . . , zs,N ) the grid (or N-tuple in Rq ) at step s, the stochastic gradient descent procedure is recursively defined by : z(s+1) = z(s) − δs+1 KN (z(s) , ξs+1 ),

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where (ξs )s are independent copies of Z, and (δs )s is a positive sequence of step parameters satisfying the usual conditions : δs = ∞ and δ2s < ∞. s

s

In our context, this leads to the Kohonen algorithm or competitive learning vector quantization (CLVQ) algorithm, which also provides as a byproduct an estimation of the weights pˆ i of the Voronoi tesselations associated to the stationary quantizer. We refer to [10] for a complete description and discussion of the convergence of algorithm. Optimal grids and their companion parameters, i.e. weights of the Voronoi tesselation and distorsion, for the normal distribution are available and downloadable on the webpages of Gilles Pagès or Jacques Printems. 5. Quantization of the Filter Process In view of solving dynamic optimization problems under partial observation, we need an approximation of the filter process (Πk )k . Recall the dependence of the random filter on the observation : Πk = Πk (Y1 , . . . , Yk ). An usual approach, suggested e.g. in [3], consists of approximating Πk (Y1 , . . . , Yk ) by Πk (Yˆ 1 , . . . , Yˆ k ) where Yˆ k is a quantizer of Yk . The main problem in effective implementation is the growing dimension of this approximating filter : indeed, for instance, if each Yˆ k takes M values, then at time n, the random filter Πn (Yˆ 1 , . . . , Yˆ n ) would take Mn values in Km , which is not realistically implementable for a long horizon n. In order to overcome this numerical difficulty, we present a quantization approach introduced in [11] and based on the Markov property of the pair filter-observation (Πk , Yk ) with respect to the observation filtration (FkY ). In other words, the conditional law of Xk+1 given FkY is summarized by the sufficient statistic (Πk , Yk ), and we shall approximate the pair Markov chain (Πk , Yk ) by an approximation of their successive probability transitions. One first proves that the probability transition Rk (from time k − 1 to k) of the Markov chain (Zk ) = (Πk , Yk ) in Km × Rd is given by : ¯ k (π, y, y ), y )Qk (π, y, dy), Rk ϕ(π, y) = ϕ(H where Qk (π, y, dy) is the law of Yk conditional on (Πk−1 , Yk−1 ) = (π, y) with density : (5.1)

y −→

m i, j=1

ij

gk (xi , y, x j , y )Pk πi .

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This shows in particular that Zk may be simulated through the following simulation procedure of its probability transition (Rk ) : for k = 0, Z0 is a known deterministic vector equal to z0 = (µ, y0 ), and for k ≥ 1, starting from (Πk−1 , Yk−1 ), • we simulate Yk according to the law Qk (Πk−1 , Yk−1 , dy ) given in (5.1). • we compute Πk by the forward filtering equation Πk = H¯ k (Πk−1 Yk−1 , Yk ). Once we are able to simulate independent copies of (Z0 , . . . , Zn ), we apply an optimal quantization to each Zk in Km × Rd , for k = 0, . . . , n, following the vector quantization method described in the previous section. For each k = 0, . . . , n, we denote by Zˆ k the zk -Voronoi quantizer of Zk , valued in the k grid zk = (z1k , . . . , zN ) consisting of Nk points in Km × Rd associated to the k Voronoi tesselations Ci (zk ), i = 1, . . . , Nk . As a byproduct, we approximate the probability transitions (Rk ) of the Markov chain (Zk ) by the probability transition matrices (ˆrk ) defined by :

ij j rˆk = P Zˆ k = zk Zˆ k−1 = zik−1 ij P Zk ∈ C j (zk ), Zk−1 ∈ Ci (zk−1 ) βˆk = =: i , P [Zk−1 ∈ Ci (zk−1 )] pˆk−1 for all k ≥ 1, i = 1, . . . , Nk−1 , j = 1, . . . , Nk . The process (Zˆ k ) obtained by this method, is called a marginal quantization of the process (Zk ) : it is characterized for each k by its grid space z k , and by the probability ij transition matrix rˆk = (ˆrk ). Denoting by ξs = (ξs0 , . . . , ξsn )s , independent copies of (Z0 , . . . , Zn ), the optimal grids zk that minimize the L2 -quantization error Zk − Zˆ k 2 for ij each k, and the companion parameters rˆk , are practically implemented according to the Kohonen algorithm as follows : Initialisation phase : (0)

k , . . . , z0,N ) ∈ (Km × Rd )Nk for k = 0, . . . , n, • Initialize the n grids zk = (z0,1 k k

with Γ(0) = z0 reduced to N0 = 1 point for k = 0. 0 0,i j

• Initialize the weights vectors : p0,i = 1/Nk , βk+1 = 0, i = 1, . . . , Nk , j = k 1, . . . , Nk+1 , and the distorsion D0N = 0, for k = 0, . . . , n. k

(s)

k , . . . , zs,N ), the weights Updating s → s + 1 : At step s, the n grids z k = (zs,1 k k

s,i j

vectors ps,i , βk+1 , i = 1, . . . , Nk , j = 1, . . . , Nk+1 , have been obtained and we k

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use the sample ξs+1 of (Z0 , . . . , Zn ) to update them as follows : for all k = 0, . . . , n, • Competitive phase : select ik (s + 1) ∈ {1, . . . , Nk } such that ξs+1 ∈ Cik (s+1) (z(s) ), i.e. ik (s + 1) ∈ argmin1≤i≤Nk |zs,i − ξs+1 |2 . k k • Learning phase : Updating of the grid :

s,i s,i s+1 = z − δ 1 − ξ zs+1,i z , s+1 i=i (s+1) k k k k

i = 1, . . . , Nk

Updating of the weights vectors and of the probability transition

= ps,i − δs+1 ps,i − 1i=ik (s+1) , ps+1,i k k k

s,i j s+1,i j s,i j βk+1 = βk+1 − δs+1 βk+1 − 1i=ik (s+1), j=ik+1 (s+1) , s+1,i j

s+1,i j rk+1

=

βk+1

ps+1,i k

,

for all i = 1, . . . , Nk , j = 1, . . . , Nk+1 . 6. Numerical Approximation to Optimization Problems under Partial Observation 6.1 Quantization of optimal stopping We turn back to the optimal stopping problem under partial observation considered in paragraph 3.1, and we define the corresponding values : (6.1) Uk = ess sup E h(τ, Xτ , Yτ )| FkY , k = 0, . . . , n, Y τ∈Tk,n

Y where Tk,n is the set of (FkY )-stopping times valued in {k, . . . , n}. By using the law of iterated conditional expectation and the definition of the filter, we notice that problem (6.1) may be reduced to a complete observation model with state variable the (FkY )-adapted process (Zk ) : n

1τ= j E[h(j, X j , Y j )|F jY ]FkY Uk = ess sup E Y τ∈Tk,n

j=k

n

= ess sup E 1τ= j Π j h(j, ., Y j )FkY Y τ∈Tk,n

j=k

˜ Zτ )F Y , = ess sup E Πτ h(τ, ., Yτ )| FkY = ess sup E h(τ, k Y τ∈Tk,n

Y τ∈Tk,n

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with the notation : ˜ z) = πh(., y) = h(k,

m

h(k, xi , y)πi ,

∀z = (π, y), π = (πi )i ∈ Km , y ∈ Rd .

i=1

By the (FkY )-Markov property of (Zk ) and the dynamic programming principle, we have Uk = uk (Zk ) where functions uk are defined in backward induction by : ˜ z) un (z) = h(n, ˜ z) , E [ uk+1 (Zk+1 )| Zk = z] . uk (z) = max h(k, Following [1], we provide a quantization approximation of Uk = uk (Zk ) ˆ k = uˆ k (Zˆ k ), for k = 0, . . . , n, where (Zˆ k ) is a marginal quantization of by U (Zk ) on grids (zk ) with corresponding probability transition matrices (ˆrk ), as described in the previous section, and functions uˆ k are explicitly computed in recursive form by : ˜ z) uˆ n (z) = h(n, ˜ z) , E uˆ k+1 (Zˆ k+1 ) Zˆ k = z . uˆ k (z) = max h(k, From an algorithmic viewpoint, this reads as : ˜ zi ), i = 1, . . . , Nn uˆ n (zin ) = h(n, ⎧n ⎫ ⎪ ⎪ N k+1 ⎪ ⎪ ⎪ ⎪ ⎨˜ ⎬ ij j i ˆ ˆ h(k, z , ) , (z ) uˆ k (zik ) = max ⎪ r u ⎪ k+1 k+1 ⎪ k ⎪ k+1 ⎪ ⎪ ⎩ ⎭ j=1

i = 1, . . . , Nk , k = 0, . . . , n − 1.

ˆ k 1 in terms of quantization error Zk − Zˆ k 2 L1 -error estimation Uk − U is stated in [11]. By combining with Zador’s theorem, we obtain a rate of C(n) convergence of order , where C(n) is a constant depending essentially 1 N m−1+d

on the boundedness and Lipschitz conditions on gk and h, and the horizon n. Numerical illustration : Bermudean options in a partially observed stochastic volatility model We consider an observable stock (logarithm) price Yk = ln Sk , with dynamics given by : (6.2)

√ 1 Yk+1 = Yk + r − Xk2 δ + Xk δεk+1 , 2

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Table 1 Comparison of quantized filter value to its Monte Carlo estimation. Monte Carlo Quant. with N¯ = 300 Quant. with N¯ = 600 Quant. with N¯ = 900 Quant. with N¯ = 1200 Quant. with N¯ = 1500

E[Π1n ] 0.287608 0.301651 0.301604 0.301598 0.301618 0.301605

E[Π2n ] 0.422833 0.421725 0.421458 0.421316 0.42122 0.421205

E[Π3n ] 0.289558 0.276624 0.276938 0.277086 0.277162 0.27719

Relative error (%) 0.898 0.886 0.881 0.879 0.878

where (εk ) is a sequence of Gaussian white noise, and (Xk ) is the unobservable volatility process. δ = n1 is the time step from an Euler scheme over a period [0, 1]. We assume that (X k ) is a Markov chain approximation a` la Kushner [8] with spatial step ∆ and with m = 3 states of a mean-reverting process : (6.3)

dXt = λ(x0 − Xt )dt + ηdWt .

In this context of a partially observed stochastic volatility model, we consider a Bermudean put option with payoff y → (κ − e y )+ , and with price :

(6.4) . u0 = sup E e−rτδ κ − eYτ Y τ∈T0,n

+

We perform numerical tests with : - Price and put option parameters : r = 0.05, S 0 = 110, κ = 100, - Volatility parameters : λ = 1, η = 0, 1, ∆ = 0, 05, X 0 = 0.15, - Quantization : Grids are of same size N¯ fixed for each time period. We first compare in Table 1 the filter expectation at the final date computed with a time step size δ = 1/5 and by using the optimal quantization method with increasing grid size N¯ , and with 106 Monte Carlo iterations of the path observation Y. We observe that besides the very low error level, the absolute error (plotted in Fig. 1) and the relative error are decreasing as the grid size grows. Secondly, in order to illustrate the effect of the time step, we compute the American option price under partial observation when the time step δ decreases to zero (i.e. n increases) and compare it with the American option price with complete observation of (Xk , Yk ). Indeed, in the limit for δ → 0 we fully observe the volatility, and so the partial observation price should converge to the complete observation price.

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288 3

1.355

x 10

1.35

1.345

1.34

1.335

1.33

1.325

1.32

1.315 200

400

600

800

1000

1200

1400

1600

Figure 1. Filter error convergence as N¯ grows.

11

Quadratic risk

10

9

8

7

6 Total observation Partial observation 5 2.5

3

3.5

4 4.5 Initial capital

5

5.5

6

Figure 2. Quadratic hedging of an European put: graph of w 0 → infα∈A E((κ − eYn )+ − Wn )2 ) in the partial and total observation case. Size grid for W = 100 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points.

Moreover, when we have more and more observations, the difference between the two prices should decrease and converge to zero. This is shown in figure 6, where we performed option pricing over grids of size N¯ Π,Y = 1500 in case of partial observation. The total observation price is given by the same pricing algorithm carried out on N¯ X,Y = 45 points for the product grid of (Xk , Yk ). For fixed n, the rate of convergence for the

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Table 2 American option price for embedded filtrations—First Example. n

4

8

16

1.45863

1.75689

1.77642

0.921729

1.13898

1.47089

0.61

0.30

Tot. Obs. (N¯ X,Y = 30) Part. Obs. (N¯ = 1000) Variation

0.53

Π,Y

Table 3 American option price for embedded filtrations—Second Example. n

5

10

20

Tot. Obs. (N¯ X,Y = 45) Part. Obs. (N¯ = 1500)

1.57506

1.72595

1.91208

0.988531

1.30616

1.59632

Variation

0.58

0.42

0.31

Π,Y

approximation of the value function under partial observation is of order 1/(m−1+d) where N¯ Π,Y is the number of points used at each time k for the N¯ Π,Y grid of (Πk , Yk ) valued in Km × Rd . From results of [1], we also know that the rate of convergence for the approximation of the value function under full observation is of order m × N¯ Y where N¯ X,Y = m × N¯ Y is the number of points at each time k, used for the grid of (Xk , Yk ) valued in E × Rd . This explains why, in order to have comparable results, and with m = 3 and d = 1/3 . 1, we have chosen N¯ Y ∼ N¯ Π,Y In addition, it is possible to observe the effect of information enrichment as the time step decreases. In fact, if we consider multiples of n as the time step parameter, we notice that the American option price increases for both total and partial observation models (see Tables 2 and 3). 6.2 Quantization of control problem We turn back to the control problem under partial observation considered in paragraph 3.2. By using the law of iterated conditional expectations, we can rewrite the expected cost function as follows: J(α) = E E (Xn , Yn , Wn )|FnY ⎡ m ⎤ ⎢⎢ ⎥⎥ i i (x , Yn , Wn )Πn ⎥⎥⎥⎦ = E ⎢⎢⎢⎣ i=1 ˆ n , Yn , Wn ) = E (Π

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where ˆ y, w) := (π,

m

(xi , y, w)πi

i=1

The original control problem (3.2) can now be reformulated as a problem under full observation with state variables (Πk , Yk , Wk ), valued in Km × Rd × R, and (FkY )-adapted :

ˆ n , Yn , Wn ) . Jopt = inf E (Π α∈A

Recalling the dynamics (3.1) of (Wk ) and following the dynamic programming principle for discrete-time control problems, we define the sequence of functions on Km × Rd × R : ˆ y, v) un (π, y, w) = (π,

uk (π, y, w) = inf E uk+1 (Πk+1 , Yk+1 , F(w, a, y, Yk+1))(Πk , Yk ) = (π, y) , a∈A

for k = 0, . . . , n − 1, so that Jopt = u0 (µ, y0 , w0 ), where w0 is the initial value of W0 at time k = 0, and we recall that (Π0 , Y0 ) = (µ, y0 ). In order to compute this sequence of functions uk , we deal separately with the approximation of the pair filter-observation process (Zk )k = (Πk , Yk )k that does not depend on the control, and the approximation of the controlled process (Wk )k . • We apply a marginal quantization of the process (Zk ) = (Πk , Yk ), and we ˆ k , Yˆ k ) the corresponding quantizers on grids (zk ), and denote the (Zˆ k ) = (Π (ˆrk ) the associated probability transition matrices, as described in section 5. The i-th point of the grid zk of size Nk in Km × Rd is denoted zik = (πk (i), yik ) ∈ Km × Rd , i = 1, . . . , Nk . • The approximation of Wk is obtained by a classical uniform space discretization similar to the Markov chain method as in Kushner. We fix a bounded uniform grid on the state space R for the controlled process (Wk ). Namely, we set Γ = (2ν)Z ∩ [−L, L], where ν is the spatial step and L is the grid size. We denote by ProjΓ the projection on the grid Γ according to the closest neighbor rule. Recalling the dynamics (3.1) of the controlled process (Wk ), we approximate it as follows :

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291

ˆ k ), given a control α ∈ A, we define the discretized controlled process (W valued in Γ, by : ˆ k , αk , Yˆ k , Yˆ k+1 )). ˆ k+1 = Proj (F(W W Γ We then approximate the sequence of functions uk by the sequence of functions uˆ k defined on zk × Γ, k = 0, . . . , n, by a dynamic programming type formula : ˆ y, w) uˆ n (π, y, w) = (π,

ˆ ˆ ˆ ˆ ˆ uˆ k (π, y, w) = inf E uˆ k+1 Πk+1 , Yk+1 , ProjΓ (F(w, a, y, Yk+1)) (Πk , Yk ) = (π, y) . a∈A

From an algorithmic viewpoint, this is computed explicitly as follows : ˆ in , w), zin = (πn (i), yin ) ∈ zn , i = 1, . . . , Nn , w ∈ Γ, uˆ n (zin , w) = (z N k+1

j ij j rˆk+1 uˆ k+1 zk+1 , ProjΓ (F(w, a, yik, yk+1 )) (6.5) uˆ k (zik , w) = inf a∈A

(6.6)

zik

j=1

= (πk (i), yik ) ∈ zk , i = 1, . . . , Nk , w ∈ Γ, k = 0, . . . , n − 1.

For w0 ∈ Γ, the solution Jopt = u(µ, y0 , w0 ) to our control problem is then approximated by Jquant = uˆ 0 (µ, y0 , w0 ). Moreover, this backward dynamic programming scheme allows us to compute at each time k = 0, . . . , n − 1, an approximate control αˆ k (z, w), z ∈ zk , w ∈ Γ, by taking the infimum in (6.5). Error estimation between Jopt and Jquant in terms of the quantization errors Zk − Zˆ k 2 for Zk = (Πk , Yk ), the spatial step ν, and the grid size L for (Wk ) is stated in [5]. By combining

with Zador’s theorem, this provides a rate of convergence of order C(n) ν + L1 + 11 . N m−1+d

Numerical illustration : Mean-variance hedging in a partially observed stochastic volatility model In the setting of the stochastic volatility model described in paragraph 6.1, we consider the mean-variance hedging of a put option. The logarithm of the observed stock price is Y = ln S, its unobservable volatility is X, and the wealth process W controlled by the number of shares α invested in stock, is governed by : Wk+1 = Wk erδ + αk (eYk+1 − eYk erδ ), where r is the constant interest rate, and δ > 0 is the interval between two trading dates. The dynamics of (X, Y) is given by (6.2)-(6.3). Given a put

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300 points 600 points 1500 points

8.7 8.6 8.5 8.4 8.3 8.2 8.1 8 7.9 7.8 2

2.5

3

3.5

4

Figure 3. Quadratic hedging of an European put: graph of w 0 → infα∈A E((κ − eYn )+ − Wn )2 ) for different quantification grid sizes (N = 300, 600, 1500) and a fixed uniform grid size (N W = 400)

Table 4 Quadratic hedging of an European put: European put price (defined as the initial capital minimizing the risk) and optimal control strategy calculated for different quantization grid sizes (N = 300, 600, 1500) and a fixed uniform grid size (NW = 400) N 300 600 1500

European put price 3.04132 3.05965 3.07098

Optimal control strategy α0 -0.2813 -0.2813 -0.2813

option of payoff (κ − eYn )+ at maturity n, the investor’s objective is defined by the control problem : 2

inf E (κ − eYn )+ − Wn .

α∈A

We perform numerical tests with : - Price and put option parameters : r = 0.05, S 0 = 110, κ = 110, - Volatility parameters : λ = 1, η = 0, 1, ∆ = 0, 05, X 0 = 0.15, - Quantization of (Zk ) = (Πk , Yk ) : grids are of same size N fixed for each time period with step δ = n1 . When it is not precised, we choose n = 5. - Discretization of (Wk ) : we use a N W -point grid defined by Γ = (2ν)Z ∩ [Lin f , lsup ] with Lin f = −10, Lsup = 15 and so ν = 2(N25 W −1) .

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293 8.6

400 points 200 points 100 points

8.5 8.4 8.3 8.2 8.1 8 7.9 7.8 7.7 2.4

2.6

2.8

3

3.2

3.4

3.6

Figure 4. Quadratic hedging of an European put: graph of w 0 → infα∈A E((κ − eYn )+ − Wn )2 ) for different fixed uniform grid sizes (NW = 50, 100, 200, 400) and a fixed quantization grid size (N = 300)

- Approximation of the optimal control: golden search method (see [9]) on A = [−1, 1]. In order to study the effects of the quantization grid size N and uniform grid size N W , we plot the graph of w0 → infα∈A E((κ − eYn )+ − Wn )2 ) for different values of N and N W (Figs. 3 and 4). As expected, the global shape of the graph is parabolic, due to the quadratic hedging criterion that we have used. The minimum is reached at wmin which can be considered as the ”quadratic hedging price” of our European put option. The corresponding hedging strategies are given in Table 4, and Fig. 5 displays the graph of α0 as a function of the initial wealth w0 . We can see that the strategy is nearly constant for w0 ∈ [2, 4], where the non constant values may be due to numerical imprecision. This is consistent with the theoretical result, which shows that the optimal strategy for the mean-variance hedging problem does not depend on the initial wealth when the (discounted) stock price is a martingale, which is the case here. In Fig. 6 and in the Table 5, we compare the European put option price under partial and complete observation when we increase the number of observations (i.e. the time step δ decreases to zero). Denoting by NΠ,Y the number of grid points used in the partial observation case to make an optimal quantization of the pair (Π, Y), by NX,Y the number of grid points used in the total observation case to make an optimal quantization of the

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Figure 5. Quadratic hedging of an European put: graph of w 0 → α0 (w0 ) for a quantization grid size of N = 300 and a fixed uniform grid size of NW = 400

0.25

a2

0.15

0.1

0.05

0

5

10

15

20

25

Figure 6. Quadratic hedging of an European put: distance between total and partial observation European put prices (defined as the initial capital minimizing the risk) when we increase the number of observations (axis of abscissae) and consequently the time step δ goes to 0. Size grid for W = 30 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points

pair (X, Y), and by L the grid size in the discretization of the controlled variable W, we recall that the discretization error is of order −1 1 d+m−1 NΠ,Y +ν+ L

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Table 5 Quadratic hedging of an European put: comparison between partial and total observation price (defined as the initial capital minimizing the quadratic risk) and strategies when we increase the number of observations and consequently the time step δ goes to 0. Size grid for W = 30 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points Time step Partial observation Partial observation Total observation Total observation δ price strategy price strategy 1\5 2.9933 −0.2813 3.24459 −0.2734 1\10 3.5255 −0.3013 3.65515 −0.2422 1\20 3.9501 −0.3215 4.02799 −0.3614

2.5

2

1.5

1

0.5 Tot Obs Option Price (45 pts) Part Obs Option Price (1500pts)

0

0

5

10

15

20

25

30

35

Figure 7. Partial and total observation option prices as δ → 0

for the partial observation case. For the total observation case we have: ! 1 1 +ν+ NX,Y R where NX,Y = mNY (see [11]). So, in order to obtain comparable results, given the uniform grid discretizing the variable W, we perform an optimal quantization of (Π, Y) and (X, Y) by using grid sizes NΠ,Y and NX,Y = mNY such that: 1 d+m−1 NY NΠ,Y where d = 1 and m = 3. That is why we have chosen NΠ,Y = 1500 and NX,Y = 45.

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We notice that when the number of observations increases (i.e. δ → 0), the partial observation price converges to the complete observation price; this is due to the fact that with observation performed in continuous time we are able to calculate the volatility given by the quadratic variation of the price process (eY ). Figure 2 shows that by working in a total observation setting the quadratic risk associated to a given initial wealth is smaller than the corresponding value obtained in the partial observation case. This is consistent with the fact that the filtration generated by the observation price is included in the full information filtration, and consequently the corresponding optimal cost function in the partial information case is larger than the one in the full information case.

References 1. Bally, V. and G. Pagès (2003): “A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems”, Bernoulli, 9, 1003–1049. 2. Bensoussan, A. (1992): Stochastic control of partially observable systems, Cambridge university Press. 3. Bensoussan A. and W. Runggaldier (1987): An approximation method for stochastic control problems with partial observation of the state: a method for constructing ε-optimal controls, Acta Appli. Math., 10, 145–170. 4. Bouchard, B., I. Ekeland, and N. Touzi (2004): “On the Malliavin approach to Monte Carlo approximation of conditional expectations”, Finance and Stochastics, 8, 45–71. 5. Corsi, M., H. Pham, and W. Runggaldier (2006): “Numerical approximation by quantization of control problem in finance under partial observations”, to appear in Mathematical modelling and numerical methods in finance, edited by A. Bensoussan and Q. Zhang, special volume of Handbook of numerical analysis. 6. Di Masi, G. B. and W. J. Runggaldier (1987): An Approach to Discrete-Time Stochastic Control Problems under Partial Observation, SIAM J. Control & Optimiz. 25, pp. 38 - 48. 7. Graf, S. and H. Luschgy (2000): Foundations of quantization for random vectors, Lecture Notes in Mathematics n0 1730, Springer, Berlin, 230 pp. 8. Kushner, H. J. and P. Dupuis (1992): Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York. 9. Luenberger, D. (1984): Linear and nonlinear programming, Addison-Wesley. 10. Pagès, G., H. Pham, and J. Printems (2004): “Optimal quantization methods and applications to numerical problems in finance”, Handbook of computational and numerical methods in finance, ed. S. Rachev, Birkhauser. 11. Pham, H., W. Runggaldier, and A. Sellami (2005): “Approximation by quanti-

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