Control Theory and Related Topics: In Memory of Prof Xunjing Li, Fudan University, China, 3-5 June 2005

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Control Theory and Related Topics

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edited by

Shanjian Tang Fudan University, China

Jiongmin Yong University of Central Florida, USA

Control Theory and Related Topics

In Memory of Xunjing Li Fudan University, China

3 – 5 June 2005

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.

CONTROL THEORY AND RELATED TOPICS IN MEMORY OF PROFESSOR XUNJING LI (1935–2003) 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-582-2 ISBN-10 981-270-582-1

Printed in Singapore.

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Xunjing Li (1935–2003)

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PREFACE

In the past several decades extensive research has been devoted to the mathematical control theory, in particular for distributed parameter and stochastic systems, and its application in mathematical finance. Professor Xunjing Li, a distinguished educator and researcher, played a special leading role in the development of these research efforts, particularly related to the maximum principle for optimal controls in infinite dimensional state space from the late seventies of the last century until his untimely death in February, 2003. This commemorative volume collects research articles devoted to reviewing the state of the art of this and other related rapidly developing research and to exploring new directions of research in these fields. It is a tribute to the Life and Work of Professor Xunjing Li by his students, friends, and colleagues whose personal and professional lives he has deeply touched through his generous insights and dedication to his profession. During June 3–5, 2005, Fudan University organized in Shanghai the “Workshop on Control of Distributed Parameters and Stochastic Systems in Memory of Professor Xunjing Li’s Seventieth Birthday”. We would like to thank all the participants for their presence. We also thank our colleague Yuan Zhou, our post doctors Juan Li and Mingyu Xu, and our graduate students Yashan Xu, Xiaobo Bao, Lei Wang, and Liang Zhu for their efficient services for the workshop. On behalf of all the participants, we also thank Jianxiong Huang for organizing a very nice excursion to Nanxun, Huzhou. Many of the contributors in this volume are speakers at the workshop. There are a few others as well. Due to various reasons, we were not able to invite all the students, colleagues and co-authors of Professor Xunjing Li, which we feel very regret. The topics covered include several aspects of linear quadratic optimal control of deterministic and stochastic systems, controllability of stochastic and/or parabolic systems, nonlinear observers and stabilization, dynamical systems for PDEs, the maximum principle for optimally controlled quasi-linear elliptic obstacle problems, optimization of Navier-Stokes equation, assets and insurance pricing, and applied BSDEs and reflected BSDEs in the nonlinear probability. It took concerted and collective efforts from many people to produce this volume. We would like to take this opportunity to express our gratitude to them. Our sincere thanks first go to all the contributors for their contributions and for their cooperation, and, in particular, to Jin Ma and Yuncheng You for their thoughtful article: “A Tribute in Memory of Professor Xunjing Li on His Seventieth Birthday.” Our special thanks also go to the World Scientific staff and, in particular, to Editor vii

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Ms E. H. Chionh and Kim Wei Lee for their cooperation and kind patience in bringing out this volume in a timely fashion. Finally, we would like to take this opportunity to recognize the special role played by Liwan Liu, Professor Xunjing Li’s wife, whose long term support and influence to Professor Xunjing Li was extremely significant. We express to her our special gratitude for her presence at the workshop. The publication of this volume is supported by the National Basic Research Program of China (973 Program) with Grant No. 2007CB814904, which is greatly appreciated.

June 30, 2007 Shanjian Tang Jiongmin Yong

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Chap8-Contents

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Conference Photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Part One: Xunjing Li’s Academic Life A Tribute in Memory of Professor Xunjing Li on His Seventieth Birthday . . . . . . .3 Jin Ma and Yuncheng You Publications of Xunjing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Students and Post-doctors Advised by Xunjing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part Two: Stochastic Control, Mathematical Finance, and Backward Stochastic Differential Equations Axiomatic Characteristics for Solutions of Reflected Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Xiaobo Bao and Shanjian Tang A Linear Quadratic Optimal Control Problem for Stochastic Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Shuping Chen and Jiongmin Yong An Additivity of Maximum Expectations and Its Applications . . . . . . . . . . . . . . . . .67 Zengjing Chen, Matt Davison, Mark Reesor, and Ying Zhang Stochastic Control and BSDEs with Quadratic Growth . . . . . . . . . . . . . . . . . . . . . . . . 80 Marco Fuhrman, Ying Hu, and Gianmario Tessitore A Fundamental Theorem of Asset Pricing in Continuous Time with Square Integrable Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Hanqing Jin and Xun Yu Zhou

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Indifference Pricing of Universal Variable Life Insurance . . . . . . . . . . . . . . . . . . . . . . 107 Jin Ma and Yuhua Yu gΓ -Expectations and the Related Nonlinear Doob–Meyer Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Shige Peng and Mingyu Xu Nonconvexity Phenomenon on Itˆ o’s Integrals and on Stochastic Attainable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Shanjian Tang Unique Continuation and Observability for Stochastic Parabolic Equations and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Xu Zhang Part Three: Deterministic Control Systems Design of Dynamic High-Gain Observers for a Class of MIMO Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Hao Lei, Jianfeng Wei, and Wei Lin Some Counterexamples in Existence Theory of Optimal Control . . . . . . . . . . . . . . 183 Hongwei Lou A Generalized Framework for Global Output Feedback Stabilization of Inherently Nonlinear Systems with Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Jason Polendo and Chunjiang Qian A Linear Quadratic Constrained Optimal Feedback Control Problem . . . . . . . . . 215 Yashan Xu On Finite-Time Stabilization of a Class of Nonsmoothly Stabilizable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bo Yang and Wei Lin The Algebraic Criterion for Delay-Independent Stability of Linear Systems . . . 241 Xin Yu, Chao Xu and Kangsheng Liu Transformation Method of Solving a Time-Optimal Control Problem with Pointwise State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Shangwei Zhu and Xunjing Li

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Part Four: Dynamics and Optimal Control of Partial Differential Equations Optimal Control of Quasilinear Elliptic Obstacle Problems . . . . . . . . . . . . . . . . . . . 263 Qihong Chen and Yuquan Ye Controllability of a Nonlinear Degenerate Parabolic System with Bilinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Ping Lin, Hang Gao, and Xu Liu Near-Optimal Controls to Infinite Dimensional Linear-Quadratic Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Liping Pan and Qihong Chen Shape Optimization of Stationary Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . 323 Gengsheng Wang, Lijuan Wang, and Donghui Yang Solution Map of Strongly Nonlinear Impulsive Evolution Inclusions . . . . . . . . . . .338 Xiaoling Xiang, Yunfei Peng, and Wei Wei Study on Repairable Series System with Two Components — A Semigroup Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Hong Xu and Dinghua Shi Finite Dimensional Reduction of Global Dynamics and Lattice Dynamics of a Damped Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Yuncheng You Recent Progress on Nonlinear Wave Equations via KAM Theory . . . . . . . . . . . . . . 387 Xiaoping Yuan


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From left: back row — Zengjing Chen, Jianxiong Huang, Liping Pan, Qihong Chen, Gengsheng Wang, Hongwei Lou, Hanzhong Wu and Xiaobo Bao. Middle row — Hang Gao, Yuan Zhou, Kangsheng Liu, Chunjiang Qian, Xu Zhang, Xun Li, Yashan Xu and Lei Wang. Front row — Xiaoling Xiang, Shanjian Tang, Shige Peng, Shuping Chen, Yuncheng You, Jin Ma, Xunyu Zhou, Wei Lin and Ying Hu.


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Chap6-Participants

LIST OF PARTICIPANTS

Xiaobo Bao School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Qihong Chen Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected] Shuping Chen Department of Mathematics, Guizhou University, Guizhou, and Department of Mathematics, Zhejiang University, Hangzhou, China [email protected] Zengjing Chen Department of Mathematics, Shandong University, Jinan, China [email protected] Hang Gao School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Ying Hu IRMAR, Université Rennes 1, Rennes, France [email protected] Jianxiong Huang Shanghai College of Electric Power, Shanghai, China [email protected] Xun Li National University of Singapore, Singapore [email protected]

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Wei Lin Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Daobai Liu Fudan Univeristy, Shanghai, China [email protected] Kangsheng Liu Zhejiang University, Hangzhou, China [email protected] Hongwei Lou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Jin Ma Department of Mathematics, Purdue University, Indiana, USA [email protected] Liping Pan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Shige Peng School of Mathematical Sciences, Fudan Univeristy, Shanghai, and School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Chunjiang Qian Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected], [email protected] Shanjian Tang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Gengsheng Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected]

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Jianbo Wang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Lei Wang Fudan Univeristy, Shanghai, China [email protected] Hanzhong Wu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xiaoling Xiang Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China [email protected] Yashan Xu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Yuncheng You Department of Mathematics, University of South Florida,Tampa, Florida, USA [email protected] Xiaoping Yuan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xu Zhang School of Mathematics, Sichuan University, Chengdu, China [email protected] Hailang Zhou Shanghai Bank, China Xun Yu Zhou Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected]

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Yuan Zhou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Daxun Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Huimin Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Liang Zhu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Shangwei Zhu Department of Applied Mathematics, Shanxi Finance & Economics University, Taiyuan, China [email protected]

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Chap7-Contributors

LIST OF CONTRIBUTORS

Xiaobo Bao School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Qihong Chen Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected] Shuping Chen Department of Mathematics, Guizhou University, Guizhou, and Department of Mathematics, Zhejiang University, Hangzhou, China [email protected] Zengjing Chen Department of Mathematics, Shandong University, Jinan, China [email protected] Matt Davison Department of Applied Mathematics, University of Western Ontario London, ON, Canada [email protected] Marco Fuhrman Dipartimento di Matematica, Politecnico di Milano, Milano, Italy [email protected] Hang Gao School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Ying Hu IRMAR, Université Rennes 1, Rennes, France [email protected]

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Hanqing Jin Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected] Hao Lei Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Ping Lin Graduate Student, School of Mathematics & Statistics, Northeast Normal University, Changchun, China [email protected] Wei Lin Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Kangsheng Liu Zhejiang University, Hangzhou, China [email protected] Xu Liu Graduate Student, Department of Mathematics, College of Science, Zhejiang University, Hangzhou and School of Mathematics & Statistics, Northeast Normal University, Changchun, China Hongwei Lou School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Jin Ma Department of Mathematics, Purdue University, Indiana, USA [email protected] Liping Pan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected]

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Shige Peng School of Mathematical Sciences, Fudan Univeristy, Shanghai, and School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Yunfei Peng Department of Mathematics, Guizhou University, Guiyang, Guizhou, China [email protected] Jason Polendo Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected] Chunjiang Qian Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, Texas, USA [email protected], [email protected] Mark Reesor Department of Applied Mathematics, University of Western Ontario London, ON, Canada [email protected] Dinghua Shi Department of Mathematics, Shanghai University, Shanghai, China Shanjian Tang School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Gianmario Tessitore Dipartimento di Matematica, Universit` a di Parma, Parma, Italy [email protected] Gengsheng Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected] Lijuan Wang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected]

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Jianfeng Wei Department of Electrical Engineering and Computer Science, Case Western Reserve University, Ohio, USA [email protected] Wei Wei Department of Mathematics, Guizhou University, Guiyang, Guizhou, China [email protected] Xiaoling Xiang Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China [email protected] Chao Xu Zhejiang University, Hangzhou, China [email protected] Hong Xu Department of Mathematics, Shanghai University, Shanghai, China Mingyu Xu School of Mathematics and System Science, Shandong University, Jinan, China [email protected] Yashan Xu School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Bo Yang Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA [email protected] Donghui Yang Department of Mathematics, Wuhan University, Wuhan, Hubei, China [email protected] Yuquan Ye Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China [email protected]

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Jiongmin Yong Department of Mathematics, University of Central Florida, Orlando, Florida, USA, and School of Mathematical Sciences, Fudan University, Shanghai, China [email protected] Yuncheng You Department of Mathematics, University of South Florida,Tampa, Florida, USA [email protected] Xin Yu Zhejiang University, Hangzhou, China [email protected] Yuhua Yu Department of Mathematics, Purdue University, Indiana, USA [email protected] Xiaoping Yuan School of Mathematical Sciences, Fudan Univeristy, Shanghai, China [email protected] Xu Zhang School of Mathematics, Sichuan University, Chengdu, China [email protected] Ying Zhang Department of Mathematics and Statistics Acadia University, Canada [email protected] Xun Yu Zhou Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, China [email protected] Shangwei Zhu Department of Applied Mathematics, Shanxi Finance & Economics University, Taiyuan, China [email protected]


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Chap9-PartOneTitle

PART ONE XUNJING LI’S ACADEMIC LIFE

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Chap10-Tribute

A TRIBUTE IN MEMORY OF PROFESSOR XUNJING LI ON HIS SEVENTIETH BIRTHDAY

JIN MA Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA E-mail: [email protected] YUNCHENG YOU Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA E-mail: [email protected]

Professor Xunjing Li passed away in February, 2003, at age 68, in Shanghai, where he had lived for about half a century, almost equivalent to all his professional life. Professor Li was known to many of his students as “Lao Ban”, which means “Boss” in English. This is not only because of his absolute authority sensed by his students, but most importantly is because of his vision in finding new directions of research, his rigorous attitude towards each and every detail in the research work, and his role as a mentor to many young mathematicians he had fostered. He has been greatly missed by all his students, colleagues, collaborators, especially in the event such as the international conferences on stochastic control, mathematical finance, and/or backward stochastic differential equations, like the one held this year (2005) in Fudan University. Professor Xunjing Li was born in Qingdao, Shandong Province, China, on June 13, 1935. He came to Shanghai in 1956, at age of 21, as a graduate student in the Department of Mathematics, Fudan University, after receiving a Bachelor degree of Mathematics from Shandong University. He spent three years for his graduate study, under the supervision of the renowned mathematician Professor Jiangong Chen, in the area of approximation theory of functions. He started his teaching and research career in 1959, when he became an assistant professor in the Department of Mathematics, Fudan University. He was promoted to Lecturer in 1962, to Associated Professor in 1980a , and to Full Professor in 1984. In 1985, he became a nationally appointed Doctoral Supervisor, and was named as a Distinguished a From

1966 to 1976, China was in the period of “Cultural Revolution”. All academic titles in universities were abolished during that period 3

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Professor of Fudan University in 1997. Professor Li retired in 2001. During the forty-five year span of his professional life, Professor Xunjing Li grew from a student to a researcher and an educator, and made ineffaceable contributions to the advances in control theory and related fields in China. He was best known for his works on Maximum Principle for optimal controls of infinite dimensional control systems, and is one of the most prominent pioneers in the area of the optimal control theory of distributed parameter systems in China. Professor Li’s another major contribution to the Chinese and the international control community was, arguably, the fact that he almost single-handedly found and fostered a stochastic control group in Fudan University, which has produced a flock of scholars who later become influential players in the area across China, Europe and North America. Professor Li’s research achievements can be chronologically summarized according to the following main periods.

1. 1959—1976. Almost as soon as Professor Li finished his graduate study and started work as an assistant lecturer in the Department of Mathematics of Fudan University, he switched his research direction from Function Theory to Ordinary Differential Equations. This was partly due to a call from the department to expand the academic areas, and as a young faculty member Professor Li enthusiastically answered. Following the leadership of Professor Fulin Jin, Professor Li spent tremendous amount of time in the teaching and research in this new area. In 1962, when he was only 27 years old, he co-authored the text book “Ordinary Differential Equations” with Professor Fulin Jin. The book has been widely used as a main text/reference book by researchers and students in China for many years. In the meantime, as a natural extension of the theory of ordinary differential equations, Professor Li began to explore the area of control theory in early 1960’s. While working on the subject of absolute stability of (finite dimensional) dynamic systems, he participated in another important service activity to the Chinese control theory community: this time was the translation of the celebrated monograph “Mathematical Theory of Optimal Processes” by L. S. Pontryagin et al. In the middle of 1960’s, when Professor Li’s research activities just started to take off, the whole direction of China took an unfortunate turn. During the ten-year period of 19661976, the normal education and scientific research were strongly discouraged and even interrupted due to the “Cultural Revolution”. Taking the only opportunity in late 1960’s to apply his expertise in optimal control theory, Professor Li turned to industrial and applied mathematics. He participated in several projects associated with Shanghai Petroleum Refinery Factory, as well as Jin-Shan Petroleum Chemical Cooperation. These experiences later became an important factor for his perspectives towards control theory throughout his research.

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2. 1977—1987. The year 1977 marked a resurrection and a new era of Chinese education system, as well as a turning point in Professor Li’s research career. Starting from that year, many traditional teaching and research activities began to be restored, including the national college entrance examination, a long tradition in China for universities to enroll freshman students, at both undergraduate and graduate levelb . However, the scar in the scholastic activities and scientific research caused by the ten-year turmoil period was clearly visible, the research results achieved by many scholars like Professor Li but shelved for a decade became a little out-of-date. Inspired by the new spring in the scientific field, he started to ponder over the new germing points of his research. After a careful survey of articles and evaluating his strength, he decided to attack the infinite dimensional optimal control problems. The first break through came out in 1978, in a joint work with Professor Yunlong Yao, then an assistant professor. For a time optimal control problem of infinite dimensional linear systems, they realized, in general the attainable sets is not necessarily convex (a substantial difference from the finite dimensional case), but they discovered that its closure must be. Such an observation, together with the separation theorem for convex sets in infinite dimensional spaces, lead to a proof of maximum principle of time optimal control for infinite dimensional linear systems. Their work was published in the top journal in China, Scientia Sinica (“Science in China”), and was later presented in the 8th International Federation of Automation Conference (IFAC), Kyoto, Japan. While this might be considered usual by today’s standard, but back then when China was just opened up, it was indeed a highly recognizable event. The subsequent several years then witnessed a series of research accomplishments by Professor Li and his group, including the second author. The vector-valued measures in the infinite dimensional optimal control theory was investigated in depth, and the Pontryagin’s maximum principle was extended to various cases of general semi-linear evolutionary distributed-parameter systems. Among many other results, the one that involves terminal constraints is particularly worth mentioning. It was known that in the finite-dimensional case, the maximum principle requires only the differentiability of the coefficients, provided the the terminal constraint set is closed and convex. But there exist counter-examples showing that this is no longer the case in general for the infinite dimensional systems. As a consequence, seeking the proper conditions under which the maximum principle remains valid became a long-standing challenging problem. In 1985, Li and Yao successfully resolved the problem with rigor and elegance. They proved that, for the general semi-linear evolutionary distributed-parameter systems, if the terminal constraints satisfy the finite co-dimension condition, then the maximum principle holds. This result was highly recognized by the international control community, and was later regarded as the foundation of the “Fudan School” research on the infinite-dimensional optimal control theory.

b Both

authors of this article were the beneficiaries of this new policy, though at different levels.

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3. 1987—2001. Since 1985, a group of new researchers, mostly the new generation of graduate students after the class of 1977, started to join Professor Li’s research group. By 1988, some new Ph.D’s from Europe and US returned to China and became the important new addition. Among others, most notable were the Post-doctor Shige Peng (France) and Associate Professor Jiongmin Yong (USA). In 1989, Professor Li, in collaboration with Jiongmin Yong, further extended the maximum principle to the general semilinear evolutionary distributed systems with mixed initial-terminal constraints, by using Ekeland’s Variational Principle and an improved “spike” variational method. This ignited another wave of activities, and a variety of infinite dimensional versions of the maximum principle were knocked down. It is commonly recognized that finite-dimensional optimal control theory has three milestones: the maximum principle by L. S. Pontryagin, the method of dynamic programming by R. Bellman, and the linear quadratic optimal control theory by R. E. Kalman. Professor Li’s research covered essentially all the areas, although the main focus in his earlier years was on the representation of the Pontryagin maximum principle in the infinite dimensional spaces. Many of Professor Li’s works and thoughts, along with many results obtained by the control theory research group of Fudan University (a.k.a. “Fudan School”), can be found in the book “Optimal Control Theory for Infinite Dimensional Systems”, co-authored by Xunjing Li and Jiongmin Yong, and published by Birkh¨ auser in 1995. The book summarized quite exhaustively the latest results in the optimal control theory of nonlinear, deterministic, infinite dimensional systems up to that point, from the perspectives of the aforementioned three milestones. It was very well commented by researchers in the field of control theory. Although for the most part of his professional life, Professor Li considered himself a “deterministic person”, he was nevertheless in essence the main reason of the existence of several research groups, including the stochastic control group and later the mathematical finance group, in Fudan University. Since 1985, especially after he visited several universities in USA, he had a vision that Fudan had to develop the research direction on stochastic control. He began by organizing a stochastic control seminar, and directing several of his graduate students, including the first authorc , to study and to explore new problems in that area. With Professor Li’s cultivation, the scope of the research on stochastic control was quickly expanded to most of the subjects in the field. In his late years, Professor Li personally involved in many research projects on stochastic control theory. Collaborating with Shuping Chen, Ying Hu, Shige Peng, Shanjian Tang, Jionming Yong, Xunyu Zhou, and others, he worked on various problems in linear quadratic control problems, Maximum Principle for stochastic control systems with partial observations, and with jumps. Apart from these works, Professor Li also made important contributions in many other areas such as multi-player differential games, infinite dimensional c The

first author later went on to complete his Ph.D dissertation, on singular stochastic control problems, at University of Minnesota.

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linear quadratic unbounded optimal control, and optimal control of elliptic partial differential equations. As a closely related subject to his research, Professor Li also had far-reaching perspectives toward infinite-dimensional dynamical systems and applications, an area that has been rapidly and richly expanding since 1980’s. In 1982 and 1983, he invited three leading mathematicians in this area, Professors Jack Hale, George Sell, and Shui-Nee Chow, to visit Fudan University. Inspired by these successful visits and encouraged by Professor Li himself, Xiaobiao Lin (now a professor at the NCSU, Raleigh) and the second author later finished their doctoral dissertations at Brown university and at the University of Minnesota, respectively, on topics of infinitedimensional dynamical systems and then become active and prolific researchers in this area. In addition to his mathematical research, Professor Li’s life-long pedagogical achievements constitute another highlight of his life. Professor Li supervised four postdoctors, eleven Ph.D students, thirteen Master students, as well as many junior faculty members. Professor Li was famously known of being strict to his students (this in effect earned him the name “Lao Ban”, as we mentioned before). In fact, looking back, almost all his former advisees could tell some anecdotes where he or she learned lessons, sometimes embarrassing, from Professor Li. However, this might exactly be one of the main reasons that many of them became successful later on, when they became professors, researchers, supervisors, and principal investigators themselves. As one of the main figures in dynamic system and control theory in Fudan University and in China, Professor Li showed tremendous leadership by not only encouraging young faculty in his research group to boldly explore new areas, but also guiding his graduate students in their studies and investigations in areas unfamiliar to himself. This philosophy of Professor Li was the key for success in many cases with his graduate students. It was because of these efforts that the research directions of the Fudan (control theory) group expanded and developed progressively, from distributed parameter control systems in 1970’s and 1980’s, to stochastic control theory in 1980’s and 1990’s, and to mathematical finance throughout the 1990’s and continuing through the 2000’s. It would not be exaggerating to say that without Professor Li, the Fudan University would not have a control group of a history like what it is seen today. This year when we cherish the memory of Professor Xunjing Li on his seventieth birthday, we all feel that his adamant scholastic spirit and his rigorous scientific approach have more or less become a part of our own professional life in conducting research and in educating younger generations of graduate students. We are proud to be a part of “Fudan School”, and glad to see that the named of the group is being carried on by many more talented and dedicated mathematicians year after year, and hopefully for years to come. We believe that this is what Professor Li would be pleased to see as well.

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Chap10-Tribute

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Acknowledgments Another important obituary that describes Professor Xunjing Li’s role in the development of control theory and related fields in China is the preface (in Chinese) of his collection of articles1 . Our tribute can be considered an adapted version of that article, and we owe our sincere gratitude to the authors of that article—Shuping Chen, Shige Peng, and Jiongmin Yong, for their effort of collecting all the historical information regarding Professor Li’s professional life, which is indeed the basis of this tribute. References 1. Xunjing Li, Selection of Mathematical Papers by Xunjing Li, Fudan University Press, 2003.

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Chap11-PublicationsLi-ModifiedYong

PUBLICATIONS OF XUNJING LI

I. Books 1. Ordinary Differential Equations, Shanghai Science and Technology Press, 1962, 2nd ed., 1984, (with Fulin Jin et al., in Chinese). 2. Control Theory Applied to Computers, Fudan University press, 1988 (with Laixiang Sun and Yougen Chen, in Chinese). 3. Theory of Ordinary Differential Equations for Optimal Control Systems, Higher Education Press, 1989 (with Xueming Zhang and Zuhao Chen, in Chinese). 4. Optimal Control Theory for Infinite Dimensional Systems, Birkh¨ auser, 1995 (with Jiongmin Yong). 5. Basics of Control Theory, Higher Education Press, 2002 (with Jiongmin Yong and Yuan Zhou, in Chinese). 6. Collection of Mathematical Papers, Fudan University press, 2003 (forwarded in Chinese by Shuping Chen, Shige Peng and Jiongmin Yong). II. Edited Conference Proceedings 1. Control Theory of Distributed Parameter Systems and Applications, Lecture Notes in Control and Information Sciences, vol. 159, Springer-Verlag, 1991 (with Jiongmin Yong). 2. Control of Distributed Parameter and Stochastic Systems, Kluwer Academic Publishers, 1999 (with Shuping Chen, Jiongmin Yong and Xun Yu Zhou). III. Research Papers 1. Stability of second-order linear ordinary differential equations with periodical coefficients, Collection of Students’ Papers, Shandong University, 2 (1956), 1–4 (in Chinese).

9

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2. Four-point common circles and circles of invariance under bilinear transformation, Collection of Students’ Papers, Shandong University, 2 (1957), 5–9 (in Chinese). 3. An extension of Nikolisky’s theorem, Journal of Fudan University (Natural Science) no. 2 (1957), 264–269 (in Chinese, with an English summary). 4. Ces` aro summability in Banach space, Acta Mathematica Sinica, 10 (1960), 41–54 (in Chinese); English Translation in Chinese Mathematics, 2 (1960), 40–52. 5. Uniform approximation of a function by the Ces` aro means of its Faber series, Journal of Fudan University (Natural Science), 5 (1960), no. 2, 159–166 (in Chinese, with an English summary). 6. On the absolute stability of the indirect control systems Journal of Fudan University (Natural Science), 7 (1962), no. 1, 25–34 (with Huimin Xie and Junben Chen; in Chinese, with an English summary). 7. On the absolute stability of systems with time lags, Acta Mathematica Sinica, 13 (1963), 558–573 (in Chinese); English Translation in Chinese Mathematics, 4 (1963), 609–626. 8. Time optimal control of linear systems, Journal of Fudan University (Natural Science), 9 (1964), no. 4, 501–512 (with Huimin Xie; in Chinese, with a Russian summary). 9. Stability and time optimal control for automatic regulated systems, Collection of Papers, Research Institute of Mathematics, Fudan University, (1964), 95–111 (with Huimin Xie et al.; in Chinese). 10. Mathematical principle for electro-magnetic power-driven compressors (I) — resonance analysis in the case of free loads, Journal of Fudan University (Natural Science), 10 (1965), nos. 3–4, 151–154 (in Chinese). 11. Period-analysis approach to time series, Selection of Long-Term Forecast Methods, (Eastern China 1973 meeting on precipitation forecast in the flood season), Eastern China Meteorological Observatory Center, (1973), 71–73 (in Chinese). 12. Derived dynamic equation and feed-forward control for controlled plants — an application of electronic digital computers in the control of atmospheric

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distillation, Acta Math. Appl. Sinica, 1 (1976), no. 2, 33–45 (in Chinese). 13. Analytical design of control equations, Journal of Fudan University (Natural Science), 17 (1978), no. 2, 38–48 (in Chinese). 14. On the analytic design of the optimal regulator for the systems with time lags, Journal of Fudan University (Natural Science), 19 (1980), no. 2, 189– 195 (in Chinese, with an English summary). 15. An analysis of the mathematical model to the furnace of thermal dissociation for generating ethene, Journal of Fudan University (Natural Science), 19 (1980), no. 1, 115–116 (in Chinese). 16. Time optimal boundary control for systems governed by parabolic equations, Chin. Ann. Math., 1 (1980), nos. 3–4, 453–458 (in Chinese, with an English summary). 17. Time optimal control of distributed parameter systems, Scientia Sinica, 24 (1981), 455–465 (with Yao Yunlong). 18. On the evolution equation in Banach space, Chin. Ann. Math., 2 (1981), no. 4, 479–489 (in Chinese, with an English summary). 19. On applications of vector measure to the optimal control theory for distributed parameter systems, Chin. Ann. Math., 3 (1982), no. 5, 655–662 (in Chinese, with an English summary). 20. Methods and applications of control theory Nature Magazine, 5 (1982), 435– 438 (in Chinese). 21. On the stability of nonlinear control systems with time-lag, Control Theory and Applications, 1 (1984), no. 3, 117–123 (in Chinese, with English summary). 22. Vector-valued measure and the necessary conditions for the optimal control problems of linear systems, Journal of Mathematical Research and Exposition, 4 (1984), no. 4, 51–56. 23. Maximum principle of distributed parameter systems with time-lags, Springer Lecture Notes in Control & Information Science, vol. 75 (1985), 410–427 (with Yunlong Yao).

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24. Margin of stability for the optimal regulator of distributed parameter systems Control Theory and Applications, 3 (1986), no. 1, 76–82 (in Chinese, with an English summary). 25. N-person differential games governed by infinite dimensional systems, J. Optim. Theory & Appl., 50 (1986), 431–450. 26. Maximum principle of optimal periodic control for functional differential systems, J. Optim. Theory & Appl., 50 (1986), 421–429. 27. Maximum principle of optimal controls for functional differential systems, J. Optim. Theory & Appl., 54 (1987), 335–360 (with S. N. Chow). 28. On optimal control of functional differential systems, Springer Lecture Notes in Control & Information Sciences, vol. 102 (1987), 112–119 (with Fulin Jin). 29. Optimal control for infinite dimensional systems, Springer Lecture Notes in Control & Information Sciences, vol. 159 (1991), 96–105. 30. Maximum principle for optimal control of nonlinear generalized systems — finite dimensional case, Acta Automatica Sinica, 17 (1991), 17–23 (with Shige Peng; in Chinese). 31. Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control & Optim., 29 (1991), 895–906 (with Jiongmin Yong). 32. Dynamical model of neural network (I), Acta Biophysica Sinica, vol. 8, no. 2 (1992), 339–345 (with Fanji Gu and Jiong Ruan). 33. Dynamical model of neural network (II), Acta Biophysica Sinica, vol. 8, no. 3 (1992), 412–418 (with Fanji Gu and Jiong Ruan). 34. A realistic model of neural networks, J. of Electronics, vol. 9, no. 4 (1992), 289–295 (with Fanji Gu and Jiong Ruan). 35. Progress in control theory (distributed parameter systems), 1990 Yearbook of Natural Science, 3.24–3.26, Shanghai Translation Publishing Company, Shanghai, 1992 (in Chinese). 36. Optimal control theory for infinite dimensional systems, Progress in Natural Science, 2 (1992), 104–112 (with Jiongmin Yong; in Chinese).

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37. Maximum principle for optimal control problem of nonlinear generalized systems — infinite dimensional case, Acta Math. Appl. Sinica, 15 (1992), no. 1, 99–104 (with Ying Hu and Shige Peng; in Chinese with an English summary). 38. A linear quadratic optimal control problem with disturbances — an algebraic Riccati equation and differential games approach, Appl. Math. Optim., 30 (1994), 267–305 (with Shuping Chen, Shige Peng and Jiongmin Yong). 39. Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control & Optim., 32 (1994), 1447–1475 (with Shanjian Tang). 40. Maximum principle for optimal control of distributed parameter stochastic system with random jumps, Lecture Notes in Pure & Appl. Math., 152 (1994), 867–890 (with Shanjian Tang). 41. The effect of small time delays in the feedback on boundary stabilization, Science in China (Series A), 36 (1993), 1435–1443 (with Kangsheng Liu). 42. General necessary conditions for partially observed optimal stochastic controls, J. Appl. Prob., 32 (1995), 1118–1137 (with Shanjian Tang). 43. Stochastic verification theorems within the framework of viscosity solutions, SIAM J. Control & Optim., 35 (1997), 243–253 (with Jiongmin Yong and Xunyu Zhou). 44. Tracking control for nonlinear affine systems, J. Math. Control & Information, 14 (1997), 307–318 (with K. L. Teo and W. Q. Liu). 45. Stochastic linear quadratic regulator with indefinite control weight costs, SIAM J. Control & Optim., 36 (1998), 1685–1702 (with Shuping Chen and Xunyu Zhou). 46. Minimum period control problem for infinite dimensional system, Chin. Ann. Math. (Ser. B), 19 (1998), no. 1, 113–128 (with Liping Pan). 47. The equivalence between two types of exponential stabilities, Chinese Science Bulletin, 43 (Chinese Series, 1998), no. 16, 1787–1788; 43 (English Series, 1998), no. 18, 1583–1584 (with Hanzhong Wu).

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48. Linear quadratic problem with unbounded control in Hilbert spaces, Chinese Science Bulletin, 43 (1998), no. 20, 1712–1717 (with Hanzhong Wu). 49. Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation, Int. J. Control, 72 (1999), 1078–1086 (with Wei Lin). 50. Research into unexpected events: a key project of mathematical finance, Chinese Science, vol. 51, no. 2 6–9, March, 1999 (with Zhiyuan Shen and Jiongmin Yong; in Chinese). 51. Social responsibility of scientists, Chinese Science, vol. 52, no. 1, 24–25, January, 2000 (with Zhiyuan Shen; in Chinese). 52. A linear quadratic problem with unbounded control in Hilbert spaces, Differential Integral Equations, 13 (2000), no. 4–6, 529–566 (with Hanzhong Wu). 53. Necessary conditions for optimal control of elliptic systems, J. Austral. Math. Soc. (Ser. B), 41 (2000), 542–567 (with Hang Gao). IV. Conference Proceedings Papers 1. On optimal control for distributed parameter systems, Proc. of 8th IFAC World Congress, Kyoto, Japan, 1981, 207–212 (with Yunlong Yao). 2. Time optimal control of distributed Parameter systems (in Chinese), Proc. National Exchange Meeting on Control Theory and Applied Mathematics, Science Press, Beijing, 1981, 210–211 (with Yunlong Yao). 3. Some problems on optimal control theory and computer control, Proc. National Exchange Meeting on Control Theory and Applied Mathematics, Science Press, Beijing, 1981, 230–232 (in Chinese). 4. Vector-valued measure and the necessary conditions for the optimal control problems of linear systems, Proc. IFAC 3rd Symposium on Control of Distributed Parameter Systems, Toulouse, France, 1982, 503–506. 5. On the stability of nonlinear control systems with time-lags, Proc. 1983 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, China, 1986, 477–480. 6. Bounded real lemma and stability of the time lag nonlinear control systems, Proc. 9th IFAC World Congress, Budapest, Hungary, 1984, 67–71.

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7. Quadratic optimal control for generalized linear stochastic control systems, Preprint of First China-Japan Symposium on System Control & Its Appl., Hangzhou, China, 1989 (with Ying Hu and Shige Peng). 8. Robust design of regulator for controlled systems via LQ differential games, Preprint of 2nd Japan-China Joint Symposium on System control Theory & its Appl., Osaka, Japan, 1990 (with Shuping Chen). 9. Optimal control for a class of distributed parameter systems, Proc. 29th CDC, Honolulu, USA, 1990, 2319–2320 (with Jiongmin Yong). 10. Optimality conditions for controls of infinite dimensional systems, Proc. 1st World Congress on Nonlinear Analysis, Tampa, Florida, USA, August, 1992. 11. Some Advances in the Theory of Optimal Control, Proc. National Annual Meeting of Control Theory and Applications, 1992 (in Chinese). 12. On stochastic linear controlled systems, SIAM Conference on Control and Appl., Minneapolis, USA, Sept. 1992 (with Shuping Chen, Shige Peng, and Jiongmin Yong). 13. Works on Control Science of Fudan University, Proc. Annual Meeting of Shanghai Society of Automation, 1992 (with Jiongmin Yong; in Chinese). 14. Contributions to the Theory of Optimal Stochastic Controls, Differential Equations and Control Theory (Wuhan, 1994), 169-175, Lecture Notes in Pure & Appl. Math., vol. 176 (1996), Marcel Dekker (with Shanjian Tang). 15. Necessary conditions for optimal control of infinite dimensional systems, Proc. 1994 Hong Kong International Workshop on New Directions of Control and Manufacturing, Hong Kong, 1994, 214–221. 16. A Class of Non-conventional Stochastic Linear Quadratic Regulators, Proc. 2nd Asia Control Conference, 1997, 1-317-320 (with Shuping Chen and Xunyu Zhou). 17. Necessary Conditions for Optimal Controls of Infinite Dimensional Systems, Proc. 2nd Asia Control Conference, 1997, 1429–432.

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18. Global Stabilization of Nonlinear Systems with Marginally Unstable Free Dynamics by Small Controls, Proc. of 1998 American Control Conference, Philadelphia, 1998, 303–307 (with Wei Lin). 19. The Linear quadratic Optimal Control in Hilbert Spaces with Unbounded Controls, Proc. 14th World Congress of IFAC, vol. F, 1999, 121–124. 20. Optimal Control Theory: from Finite Dimensions to Infinite Dimensions, Control of Distributed Parameter and Stochastic Systems (Proc. of the IFIP WG7.2 International Conference on Control of Distributed Parameter and Stochastic Systems, June 19–22, 1998), (eds.: Shuping Chen, Xunjing Li, Jiongmin Yong and Xun Yu Zhou), Kluwer Academic Publishers, Boston, 1999, 85–94.

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Chap12-StudentsLi-ModifiedYong

STUDENTS AND POST-DOCTORS ADVISED BY XUNJING LI

I. Post-doctors. 1. Shige Peng Final research report: General Stochastic Maximum Principle, Backward Stochastic Differential Equations, and Application of Singular Perturbation in Annealing Simulation and Neutral Networks, (March, 1988—November, 1989). 2. Daode Gao Final research report: Estimation of Potential Taxation, (June, 1996—June, 1998). 3. Aiguo Kong Final research report: Research into Capital Structure of Financial Markets, (December, 1996—July, 1998). 4. Jiang Yu Final research report: Existence of Discrete Breather for a Class of Infinite Dimensional Coupled Oscillators, (June, 1999–June, 2001).

II. Doctors. 1. Shuping Chen Riccati Equation: a New Approach, and Its Applications, Ph.D. thesis, 1985. 2. Xunyu Zhou, Maximum Principle, Dynamic Programming and Their Relationship in Optimal Control Theory, Ph.D. thesis, July, 1989. 3. Ying Hu, Maximum Principle for Optimal Control of Stochastic Systems, Ph.D. thesis, July, 1989. 4. Liping Pan, Optimal Control of Distributed Parameter Systems with Time Lags, and Infinite Dimensional Leader-Follower Differential Games, Ph.D. thesis, March, 1991.

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5. Kangsheng Liu, Stabilization and Local Boundary Control of Distributed Parameter Systems, Ph.D. thesis, December, 1991. 6. Jianxiong Huang, Bifurcation and Chaos of Homoclinic Orbits in Infinite Dimensional Dynamical Systems, Ph.D. thesis, July, 1992. 7. Shanjian Tang, Optimal Control in Hilbert Space of Stochastic Systems with Random Jumps, Ph.D. thesis, January, 1993. 8. Pingjian Zhang, Differential Riccati Equations with Uncertain Coefficients and Their Applications in H ∞ -Optimization and L-Q Games, Ph.D. thesis, 1994. 9. Hang Gao, Domain Optimization and Optimal Control of Systems Governed by Elliptic Equations, Ph.D. thesis, March, 1996. 10. Hanzhong Wu, The Linear Quadratic Optimal Control Problem with Unbounded Control in Hilbert Spaces, Ph.D. thesis, July, 1998. 11. Qihong Chen, Indirect Obstacle Optimal Control Problem for Variational Inequalities, Ph.D. thesis, July, 1999. 12. Shangwei Zhu, Two Problems in Optimal Control Theory and Applications, Ph.D. thesis, July, 2005. III. Masters. 1. Yuncheng You, Optimal Control of Linear Systems in Abstract Spaces with Indefinite Quadratic Criteria, Master thesis, January, 1981. 2. Guozhu Gao, Uniform Asymptotic Stability of Functional Differential Equations of Neutral Type, Master thesis, July, 1981. 3. Jin Ma, On Infinite-Time State Estimation, Master thesis, July, 1984. 4. Yinping Wang, Convergence of Self-Tuned Regulators, Master thesis, July, 1985. 5. Huiheng Zheng, Stability of Nonlinear Control Systems with Time Lags, Master thesis, July, 1986.

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6. Yan Qin, Singular Perturbation Approach to Computation of Optimal Control Problems, Master thesis, July, 1987. 7. Yuan Zhou, Singular Quadratic Optimal Control of Infinite Dimensional Linear Systems, Master thesis, July, 1987. 8. Hong Xu, Necessary Conditions for Optimal Control of Distributed Parameter Systems, Master thesis, July, 1988. 9. Chunjiang Qian, Positive Real Lemma and Linear Quadratic Optimal Control of Time Delayed Systems, Master thesis, July, 1994. 10. Guanghui Li, The Positive Real Lemma and Absolute Stability for Neutral Differential-Difference Equation, Master thesis, July, 1994. 11. Qi Zhou, High-Order Necessary Conditions for Singular Optimal Control of Distributed Parameter Systems, Master thesis, March, 1995. 12. Shihong Wang, Averaging of Hamilton-Jacobi Equation in Infinite Dimensions and Application, Master thesis, July, 1997. 13. Zuoyi Zhou, H∞ Control Problem for Nonlinear Systems, Master thesis, July, 1997.


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Chap13-PartTwoTitle

PART TWO STOCHASTIC CONTROL, MATHEMATICAL FINANCE, AND BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

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Chap13-PartTwoTitle

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Chap14-BaoTang-WithProofs

AXIOMATIC CHARACTERISTICS FOR SOLUTIONS OF REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS∗

XIAOBO BAO Institute of Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China & Key Laboratory of Mathematics for Nonlinear Sciences, (Fudan University), Ministry of Education, China SHANJIAN TANG Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, & Key Laboratory of Mathematics for Nonlinear Sciences, (Fudan University), Ministry of Education, China E-mail: [email protected]

In this paper, we introduce the notion of an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with a floor in terms of four axioms. We show that an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t , 0 ≤ s ≤ t ≤ T } with a continuous upper-bounded floor {St , 0 ≤ t ≤ T }, is necessarily represented by the solutions of a backward stochastic differential equation reflected upwards on the floor {St , 0 ≤ t ≤ T }, if it is E µ -super-dominated for some µ > 0 and if it has the non-increasing and floor-above-invariant property of forward translation. Keywords: American contingent claims, reflected BSDE, dynamic nonlinear evaluation, filtration-consistent nonlinear expectation, nonlinear Doob-Meyer decomposition. 2000 Mathematics Subject Classification: Primary 60H10; Secondary 60H30, 60A05, 49N90, 91B30, 91B24.

1. Introduction Let {Bt , 0 ≤ t ≤ T } be a d-dimensional standard Brownian motion defined on a probability space (Ω, F, P ). Let {Ft , 0 ≤ t ≤ T } be the natural filtration of ∗ This

work is partially supported by the NSF of China under Grant No. 10325101 (distinguished youth foundation), the National Basic Research Program of China (973 program) with Grant No. 2007CB814904, and the Chang Jiang Scholars Program. 23

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{Bt , t ∈ [0, T ]}, augmented by all P -null sets of F. Set L2 (FT ) := {ξ : ξ is an FT -measurable random variable s.t. E|ξ|2 < +∞}, L2F (0, T ; Rm) := φ : φ is Rm -valued and {Ft , 0 ≤ t ≤ T }-adapted s.t. Z T E |ϕt |2 dt < ∞ , 0 n o H2 := ϕ ∈ L2F (0, T ; R) : ϕ is predictable , n o S2 := ϕ ∈ H2 : ϕ is a continuous process s.t. E max |ϕt |2 < +∞ . 0≤t≤T

2

2

2

For ξ ∈ L (FT ), denote by L (FT ; ξ) the subset of L (FT ) whose elements are not less than ξ. We shall use L2+ (FT ) to stand for L2 (FT ; 0). Consider the following reflected backward stochastic differential equation (RBSDE):  Z T Z T    Ys = ξ + g(r, Yr , Zr )dr + KT − Ks − hZr , dBr i, 0 ≤ s ≤ T; s sZ (1) T    Ys ≥ Ss , a.s.0 ≤ s ≤ T ; K0 = 0 and (Yr − Sr )dKr = 0. 0

Here, the three objects are previously given: a terminal value ξ, a random field g : Ω × [0, T ] × R × Rd −→ R, and a continuous progressively measurable real-valued random process {St , 0 ≤ t ≤ T }. Assume that (C1) ξ ∈ L2 (FT ). (C2) g(·, y, z) ∈ H2 for (y, z) ∈ R × Rd . (C3) |g(t, y, z) − g(t, y 0 , z 0 )| ≤ α(|y − y 0 | + |z − z 0 |) a.s. with y, y 0 ∈ R and 0 z, z ∈ Rd for some positive constant α. And (C4) S + ∈ S2 and ST ≤ ξ a.s.. The solution to RBSDE (1) is a triple {(Yt , Zt , Kt ), 0 ≤ t ≤ T } of {Ft , 0 ≤ t ≤ T }-progressively measurable processes taking values in R × Rd × R+ such that (i) Z ∈ H2 , and (ii) {Kt , 0 ≤ t ≤ T } is continuous and increasing. In view of El Karoui et al., 5 we know that there exists unique solution {(Yt , Zt , Kt ), 0 ≤ t ≤ T } of RBSDE (1) if the four conditions (C1)–(C4) are satisfied. Define r;g,S Et,T [ξ] := Yt ,

∀ ξ ∈ L2 (FT ; ST )

and r,g r;g,0 Et,T [ξ] := Et,T [ξ],

∀ ξ ∈ L2+ (FT ).

Here, the superscript r indicates that the underlying operators are generated by a reflected BSDE, and the superscripts (g, S) specify the generator and the obstacle of the underlying RBSDE. El Karoui and Quenez 8 argued that the operators r;g,S , 0 ≤ s ≤ t ≤ T } introduce a nonlinear pricing system for square-integrable {Es,t American contingent claims, possessing the following properties:

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(A1) Monotonicity. Es,t [Y ] ≥ Es,t [Y 0 ] if Y, Y 0 ∈ L2 (Ft ; St ) such that Y ≥ Y 0 . (A2) Time consistency. Er,s [Es,t [Y ]] = Er,t [Y ] if r ≤ s ≤ t ≤ T and Y ∈ L2 (Ft ; St ). A financial theoretically interesting problem is the converse one: Is a dynamic operator that possesses similar properties like (A1) and (A2), necessarily represented by a RBSDE? That is, we are concerned with the axiomatic characteristics for a RBSDE. Throughout the paper, we make the following assumption (denoted by (H0)): (H0) The predictable process S is continuous, S + ∈ S2 , and there is a positive constant C such that ess sup0≤t≤T St ≤ C a.s.. The process S is given, either as the (lower) obstacle of the underlying RBSDE or as the floor of the underlying dynamic operator (see Definition (1.1) below). In this paper, we shall formulate and discuss the above converse problem for RBSDE (1) with a given general upper-bounded floor S > −∞. More precisely, we introduce the notion of a dynamic operator with a floor S, and then represent the dynamic operator by a BSDE reflected upwards on the floor S. In this way, we characterize on one hand the solutions of RBSDEs in terms of axioms, and on the other hand, we give a representation for the dynamic operator in terms of RBSDEs (1). We first introduce the following notion of an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with floor S. Definition 1.1. A time parameterized system of operators Es,t [·] : L2 (Ft ; St ) −→ L2 (Fs ; Ss ),

0≤s≤t≤T

(2)

is called an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator with floor S if it satisfies the following: Es,t [Y ] is continuous in (s, Y ) ∈ [0, t] × L2 (Ft ; St ) for t ∈ [0, T ], and furthermore, it satisfies the following four axioms. (D1) Floor-above strict monotonicity Es,t [Y ] ≥ Es,t [Y 0 ] a.s. for Y and 0 Y ∈ L2 (Ft ; St ) such that Y ≥ Y 0 a.s.. If Es,t [Y ] > Ss a.s. for any s ∈ [r, t] and some r ∈ [0, t], then Y 0 = Y a.s. if Y 0 ≥ Y a.s. and Er,t [Y 0 ] = Er,t [Y ] a.s.. (D2) Es,t [Y ] = Y a.s. for each Y ∈ L2 (Fs ; C). (D3) Time consistency Er,s [Es,t [Y ]] = Er,t [Y ] a.s. for Y ∈ L2 (Ft ; C) if r ≤ s ≤ t ≤ T. ˜ − C˜ = 1A (Es,t [Y + C] ˜ − C), ˜ (D4) Zero-one law For each s ≤ t, Es,t [1A Y + C] a.s., ∀A ∈ Fs for any constant C˜ dominating the floor. The main result of the paper is stated as follows. Theorem 1.1. Consider an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t , 0 ≤ s ≤ t ≤ T } with the floor S satisfying assumption (H0). We make the following two assumptions (H1) and (H2): (H1) E µ -super-domination. There is some µ > 0 such that µ Et,T [X + Y ] − Et,T [X] ≤ Et,T [Y ], a.s.

(3)

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26 µ for t ∈ [0, T ], X ∈ L2 (FT ; ST ) and Y ∈ L2+ (FT ). Here, Et,T [Y ] := yt with {yt , 0 ≤ t ≤ T } being the first component of the solution of the following BSDE: Z T Z T ys = Y + µ|zr |dr − hzr , dBr i, 0 ≤ s ≤ T. (4) s

s

(H2) The non-increasing and floor-above-invariant property of forward translation. Et,T [X + Y ] ≤ Et,T [X] + Y , a.s. for X ∈ L2 (FT ; ST ) and Y ∈ L2+ (Ft ). Furthermore, for X ∈ L2 (FT ; ST ) such that Et,T [X] > St a.s. for t ∈ [0, T ], We have Et,T [X + Y ] = Et,T [X] + Y for any Y ∈ L2+ (Ft ). Then, there is a random field g : Ω × [0, T ] × Rd → R such that the following are satisfied: (i) g(t, 0) = 0 for a.e. t ∈ [0, T ], (ii) |g(t, z1 ) − g(t, z2 )| ≤ µ|z1 − z2 |, and r;g,S (iii) Es,t (Y ) = Es,t (Y ) for any Y ∈ L2 (Ft ; St ) and s ∈ [t, T ] with t ∈ [0, T ]. For the particular case of the zero floor (S = 0), the last assertion is still true if the assumption (H1) is replaced by the weaker one (H1)’: (H1)’ E µ -domination. There is some µ > 0 such that µ E0,T [X + Y ] − E0,T [X] ≤ E0,T [Y ], for any X ∈ L2 (FT ; ST ) and Y ∈ L2+ (FT ). (5)

Note that the assumption (H1)’ is much weaker than that of E µ -domination used by Coquet,et al., 3 in that Y is here restricted within L2+ (FT ) instead of taking values in the whole space L2 (FT ) like the latter. This difference will complicate our subsequent arguments. Remark 1.1. For the general case of the upper bounded floor S, the primal dyC namic operator {Es,t , 0 ≤ s ≤ t ≤ T } deduces the following new one {Es,t ,0 ≤ s ≤ t ≤ T} : C Es,t [·] : L2 (Ft ; St − C) −→ L2 (Fs ; Ss − C), 0 ≤ s ≤ t ≤ T

(6)

C Es,t [X] := Es,t [X + C] − C, ∀X ∈ L2 (Ft ; St − C).

(7)

with

C It is easy to prove that Es,t [X] for X ∈ L2 (Ft ; St − C), satisfies all the conditions (H1), (H2), and (D1)–(D4). Then, it follows immediately that if Theorem 1.1 is true for the negative floor S − C, it is also true for the general upper bounded floor S. Hence, it is sufficient to prove Theorem 1.1 for the case of the negative floor.

Even in the simpler case of negative floors, two key points are still worth to be mentioned here for the proof of Theorem 1.1. One is to make full use of the non-increasing and floor-above-invariant assumption (H2) of forward translation and the assumption (H1) of E µ -super-domination for some µ > 0, to extend the underlying {Ft , 0 ≤ t ≤ T }-consistent dynamic operator from the subset L2 (FT ; ST ) of floor-dominating square-integrable random variables to the whole space L 2 (FT )

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27

of square-integrable random variables. The extended dynamic operators are shown e The generator to be identified to an {Ft , 0 ≤ t ≤ T }-consistent expectation E. g of its BSDE representation given by Theorem 2.3 turns out to be that of the desired RBSDE. The second key point is the following observation: the process e {Et,T [X], 0 ≤ t ≤ T } turns out to be an E-supermartingale for each X ∈ L2 (FT ; ST ). This fact allows us to apply a nonlinear Doob-Meyer’s decomposition theorem in Ref. 3, to give the increasing process of {Et,T [X], 0 ≤ t ≤ T } as a solution to some BSDE reflected upwards on the floor. Eventually, our proof of Theorem 1.1 is both natural and elegant. To diverse the difficulty in the above arguments, the whole proof is divided into Sections 2 and 3. The above two key points are exposed in detail separately in these two sections. The main result of this work has been announced in Bao and Tang 1 . In Bao and Tang 2 , we study a dynamic operator with a very general continuous floor S which may be unbounded from the above. We define a dynamic operator as a stopping-times parameterized system of operators, and also give the representation by RBSDEs of a dynamic operator. The rest of the paper is organized as follows. In Section 2, we consider the case of the zero floor. We concentrate our attention to show how to extend the family of dynamic operators defined on the subset L2+ (FT ) of L2 (FT ) to an {Ft , 0 ≤ t ≤ T }expectation, which is defined on the whole space L2 (FT ). Section 3 is devoted to the case of the negative floor. Restricting the underlying {Et,T , 0 ≤ t ≤ T }-consistent dynamic operator to L2+ (FT ), we get an {Ft , 0 ≤ t ≤ T }-expectation by extending the restriction to L2 (FT ) in the way as shown in the preceding section. The {Ft , 0 ≤ t ≤ T }-expectation gives the generator g of an RBSDE by Theorem 2.3. In addition, we have to give the amount {KtX , 0 ≤ t ≤ T } to push upwards for {Et,T [X], 0 ≤ t ≤ T } with X ∈ L2 (FT ; ST ), that is, the increasing process in relevant to an RBSDE. For this purpose, a nonlinear Doob-Meyer’s decomposition theorems is shown how to be used. The key point is to observe that the process {Et,T [X], 0 ≤ t ≤ T } turns e out to be an E-supermartingale for each X ∈ L2 (FT ; ST ). 2. The Case of the Zero Floor In the case of the zero floor, the properties (D2) and (D4) read (D2)’ Es,t [Y ] = Y a.s. for Y ∈ L2+ (Fs ) and s ∈ [0, t]. (D4)’ Zero-one law Es,t [1A Y ] = 1A Es,t [Y ], a.s. for s ∈ [0, t], A ∈ Fs , and Y ∈ L2+ (Ft ). From (D1), (D2)’, (H1) and (H2), we can give the following: (H2)’ We have for t ∈ [0, T ], X ∈ L2+ (FT ) and Y ∈ L2+ (Ft ), Et,T [X + Y ] = Et,T [X] + Y a.s. . In fact, consider X ∈ L2+ (FT ) and Y ∈ L2+ (Ft ). It follows from (D2)’ that Et,T [] = for t ∈ [0, T ] and very constant > 0. Therefore, from (D1), we deduce that Et,T [ + X] ≥ Et,T [] = > 0 a.s. for t ∈ [0, T ]. Consequently, by (H2), we

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28

conclude that Et,T [ + X + Y ] = Et,T [ + X] + Y a.s. .

(8)

By (H1), we can take the limit → 0, and we have the desired equality. Remark 2.1. Assume that conditions (C2) and (C3) are satisfied. Moreover, assume that g(·, ·, 0) ≡ 0. Let (y, z) be the adapted solution to the following BSDE: Z T Z T Yt = ξ + g(s, Ys , Zs )ds − hZs , dBs i, 0 ≤ t ≤ T. (9) t

t

L2+ (FT ).

with the terminal condition ξ ∈ Then the triple (y, z, 0) is the adapted solution of RBSDE (1) with the parameter (Y, g, 0). This shows that the adapted solution of RBSDE (1) gives the one of the corresponding BSDE (9) for any terminal value ξ ∈ L2+ (FT ) since the obstacle {St , 0 ≤ t ≤ T } is not active in this case. The following lemma is immediate and will be used later. Lemma 2.1. We have µ (i) E Et,T [X]p ≤ exp(2−1 p(p − 1)−1 µ2 (T − t))E[X p ] for all µ > 0, t ∈ [0, T ], p and X ∈ L (FT ), with p ∈ (1, 2]; −µ µ (ii) Et,T [X + Y ] = Y − Et,T [−X] a.s. for all µ > 0, t ∈ [0, T ], X ∈ L2 (FT ), and 2 Y ∈ L (Ft ); And µ −µ (iii) E0,T [·] and E0,T [·] are strongly continuous in L2 (FT ). Proof. We only prove the first assertion (i). The other two assertions are easy to see. µ For simplicity of notations, set yt := Et,T [X]. Then, by definition, there is unique 2 d z ∈ LF (0, T ; R ) such that (y, z) is the unique adapted solution of BSDE (4). Using Itˆ o’s formula, we have Z T Z T 1 E|yt |p + p(p−1) |ys |p−2 |zs |2 ds = E|X|p +2µp |ys |p/2 |ys |p/2−1 |zs | ds. (10) 2 t t Since µp|ys |p/2 |ys |p/2−1 |zs | ≤

1 p µ2 |ys |p + p(p − 1)|ys |p−2 |zs |2 , 2(p − 1) 2

(11)

we have E|yt |p ≤ E|X|p +

p µ2 2(p − 1)

Z

T

|ys |p ds.

(12)

t

The standard arguments of using Gronwall’s inequality then gives the desired inequality. Remark 2.2. See Coquet et al. 3 for the detailed proof of the first assertion in the case of p = 2, which is easy and standard.

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29

In view of Remark 2.1, from Proposition 3.6, page 164, Lemma 3.2 and Proposition 3.7, page 165 of Ref. 12 concerning the properties of the solutions to BSDEs, we have the following Theorem 2.1. Assume that conditions (C2) and (C3) are satisfied. Moreover, r,g assume that g is independent of y and g(·, ·, 0) ≡ 0. Then, {Es,t ,0 ≤ s ≤ t ≤ T} satisfies (H1), (D1), (D2)’, (D3), (D4)’ and (H2)’. Therefore, it is an {F t , 0 ≤ t ≤ T }-consistent dynamic operator with the zero floor. Roughly speaking, Theorem 2.1 asserts that an RBSDE with the obstacle being zero introduces an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator. In what follows, we shall consider the converse problem. That is, we shall associate an {Ft , 0 ≤ t ≤ T }consistent dynamic operator with the zero floor to a BSDE reflected on the zero floor. For this purpose, we establish the following six preliminary lemmas. First, we introduce some notations. Definition 2.1. For an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t [·], 0 ≤ s ≤ t ≤ T }, define the system of operators E[·|Ft ] : L2+ (FT ) −→ L2+ (Ft ), 0 ≤ t < T by E[Y |Ft ] := Et,T [Y ], a.s. for Y ∈ L2+ (FT )

(13)

2

and the nonlinear functional E[·] : L (FT ) → R by E[Y ] := E0,T [Y ] for Y ∈ L2+ (FT ).

(14)

The two notations E[·] and E[·|Ft ] behave in L2+ (FT ) exactly like an {Ft , 0 ≤ t ≤ T }-consistent expectation and its conditional {Ft , 0 ≤ t ≤ T }-consistent expectation on Ft . The only difference lies in the domains of variables: the former’s are L2+ (FT ), while the latter’s are L2 (FT ). This can be seen from the following obvious lemma. Lemma 2.2. For an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator {Es,t [·], 0 ≤ s ≤ t ≤ T }, E[·] and E[·|Ft ] have the following properties (E1)’-(E3)’. (E1)’ E[Y1 ] ≤ E[Y2 ] if Y1 , Y2 ∈ L2+ (FT ) and Y1 ≤ Y2 . Furthermore, if Y1 ≤ Y2 , then Y1 = Y2 if E[Y1 ] = E[Y2 ]. e > 0. And (E2)’ E[C] = C for any constant C (E3)’ For any Y ∈ L2+ (FT ), there is unique η ∈ L2+ (Ft ) such that E[η1A ] = E[Y 1A ],

∀A ∈ Ft ,

(15)

which is equal to E[Y |Ft ]. Therefore, E[Y |Ft ] can be viewed as the E-expectation of Y ∈ L2+ (FT ) conditioned on Ft for any t ∈ [0, T ], though E is in general not an {Ft , 0 ≤ t ≤ T }-consistent expectation at all. Lemma 2.3. Assume that E[Y |Ft ] satisfies conditions (H1)’ for µ > 0. Then, 1 |E[ξ1 ] − E[ξ2 ]| ≤ exp( µ2 T )||ξ1 − ξ2 ||L2 , 2

∀ξ1 , ξ2 ∈ L2+ (FT ).

(16)

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30

Therefore, E[·] is strongly continuous in L2 (FT ). Proof. Using (D1) and (H1)’, we have E[ξ1 ] − E[ξ2 ] ≤ E[|ξ1 − ξ2 | + ξ2 ] − E[ξ2 ] ≤ E µ [|ξ1 − ξ2 |]. Using Lemma 2.1, we have E µ [|ξ1 − ξ2 |]2 ≤ exp (µ2 T )E[|ξ1 − ξ2 |2 ].

(17)

Therefore, 1 1 E[ξ1 ]−E[ξ2 ] ≤ exp( µ2 T )(E|ξ1 −ξ2 |2 )1/2 = exp( µ2 T )||ξ1 −ξ2 ||L2 , 2 2

∀ξ1 , ξ2 ∈ L2+ (FT ). (18)

Identically, we can show 1 E[ξ2 ] − E[ξ1 ] ≤ exp( µ2 T )||ξ1 − ξ2 ||L2 , 2 The proof is then complete.

∀ξ1 , ξ2 ∈ L2+ (FT ).

(19)

Lemma 2.4. (E µ -domination) Assume that E[Y |Ft ] satisfies (H1)’ and (H2)’. Then, we have E −µ [Y ] ≤ E[X + Y ] − E[X] ≤ E µ [Y ],

∀X, Y ∈ L2+ (FT ).

(20)

Proof. In view of (H1)’, it is sufficient to prove the following E[X + Y ] − E[Y ] ≥ E −µ [Y ],

∀X, Y ∈ L2+ (FT ).

(21)

If Y ≤ n a.s. for some integer n, then using (H2)’, (H1)’, and Lemma 2.1, we have n − (E[X + Y ] − E[X]) = n + E[X] − E[X + Y ] = E[X + n] − E[X + Y ] ≤ E µ [n − Y ] ≤ n − E −µ [Y ]. Therefore, E[X + Y ] − E[X] ≥ E −µ [Y ].

(22)

In general, consider Y ∈ L2+ (FT ). Define Yn := Y 1{Y ≤n} . Then, Yn converges to Y strongly in L2 (FT ). The above arguments show that E[X + Yn ] − E[X] ≥ E −µ [Yn ]. Then using Lemma 2.3, the desired result (21) follows by passing to the limit n → ∞ in the last inequality. Remark 2.3. The above proof of Lemma 2.4 is more complicated than that of Ref. 3 since both assumptions (H1)’ and (H2)’ are weaker.

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31 ζ For ζ ∈ L2+ (FT ), define the operator Es,t [·] : L2+ (Ft ) → L2 (Fs ) by ζ Es,t [X] := Es,t [X + ζ] − Es,t [ζ],

∀X ∈ L2+ (FT ).

(23)

Identically as in the case of an {Ft , 0 ≤ t ≤ T }-expectation (see Ref. 3 for detailed proof), we can show the following lemma. Lemma 2.5. Let ζ ∈ L2+ (FT ). If {Es,t [·], 0 ≤ s ≤ t ≤ T } is an {Ft , 0 ≤ t ≤ T }consistent dynamic operator defined on L2+ (FT ) and satisfies (H1)’ and (H2)’, then ζ the operator {Es,t [·], 0 ≤ s ≤ t ≤ T } is also an {Ft , 0 ≤ t ≤ T }-consistent dynamic operator defined on L2+ (FT ), and satisfy (H1)’ and (H2)’. The expectation E ζ [X|Ft ] of X ∈ L2+ (FT ) conditioned on Ft is given by the formula: E ζ [X|Ft ] = E[X + ζ|Ft ] − E[ζ|Ft ].

(24)

Lemma 2.6. Assume that the two F-consistent dynamic operators E 1 [·] and E 2 [·] defined on L2+ (FT ) satisfy (H1)’ and (H2)’. If E 1 [X] ≤ E 2 [X],

∀X ∈ L2+ (FT ),

then for all t, E 1 [X|Ft ] ≤ E 2 [X|Ft ],

a.s. for all X ∈ L2+ (FT ).

Proof. The proof is divided into the two steps. Step 1. The case of X ≤ n for some positive integer n. Set η = E 2 [X|Ft ] − E 1 [X|Ft ]. Then −η1{η≤0} = E 1 [X1{η≤0} |Ft ] − E 2 [X1{η≤0} |Ft ] ≥ 0 and n ≤ E 1 [−η1{η≤0} + n] = E 1 [E 1 [X1{η≤0} |Ft ] − E 2 [X1{η≤0} |Ft ] + n] = E 1 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n] ≤ E 2 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n] = E 2 [E 2 [X1{η≤0} − E 2 [X1{η≤0} |Ft ] + n|Ft ]] = E 2 [E 2 [X1{η≤0} |Ft ] + n − E 2 [X1{η≤0} |Ft ]] = n. Thus, E 1 [−η1{η≤0} + n] = n. From the floor-above strict monotonicity (D1), we have −η1{η≤0} = 0.

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32

That is, P ({η ≤ 0}) = 0,

a.s.. Thus, E 1 [X|Ft ] ≤ E 2 [X|Ft ],

a.s..

Step 2. The general case of X ∈ L2+ (FT ). For X ∈ L2+ (FT ), define the truncation Xn := X1{X≤n} . Then it follows from Lemma 2.3 that lim E 1 [Xn ] = E 1 [X].

n→∞

From Step 1, we have E 1 [Xn |Ft ] ≤ E 2 [Xn |Ft ],

n = 1, 2, . . . .

Obviously, it is sufficient to prove the following lim E 1 [Xn |Ft ] = E 1 [X|Ft ], ∀X ∈ L2+ (FT )

n→∞

(25)

We now prove it by contradiction. Otherwise, there exists 0 ≤ ≤ 1 and A ∈ Ft such that P (A) > 0 and E 1 [Xn |Ft ]1A ≤ (E 1 [X|Ft ] − )1A . Since E 1 [Xn + 1|Ft ] = E 1 [Xn |Ft ] + 1 and E 1 [X + 1|Ft ] = E 1 [X|Ft ] + 1 (by (H2)’), we have E 1 [Xn + 1|Ft ]1A ≤ (E 1 [X + 1|Ft ] − )1A . Using (D1), we have E 1 [E 1 [Xn + 1|Ft ]1A ] ≤ E 1 [(E 1 [X + 1|Ft ] − )1A ]. Then letting n → ∞, we have lim E 1 [E 1 [Xn + 1|Ft ]1A ] ≤ lim E 1 [(E 1 [X + 1|Ft ] − )1A ]

n→∞

n→∞

< E 1 [(X + 1)1A ].

(26)

While we have by Lemma 2.1 the following lim E 1 [E 1 [Xn + 1|Ft ]1A ] = lim E 1 [(Xn + 1)1A ]

n→∞

n→∞

= E 1 [(X + 1)1A ].

(27)

This contradicts (26). Therefore, (25) is true. Combining Lemmas 2.6 and 2.4, we have Lemma 2.7. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. Then, for each t ≤ T , we have E −µ [Y |Ft ] ≤ E[X +Y |Ft ]−E[X|Ft ] ≤ E µ [Y |Ft ], a.s. for all X, Y ∈ L2+ (FT ). (28) Remark 2.4. Lemma 2.7 shows that (H1)’ together with (H2)’ and (D1)–(D4) implies (H1), as pointed in the introduction.

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33

In the following, we extend the underlying {Ft , 0 ≤ t ≤ T }-consistent dynamic operator from the subset L2+ (FT ) of nonnegative square-integrable random variables to the whole space L2 (FT ) of square-integrable random variables. As a preliminary, we have the following fact. Lemma 2.8. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. Let Xn := X1{X≥−n} and Yn := E[Xn + n|Ft ] − n for X ∈ L2 (FT ) and n = 1, 2, . .p .. Then {Yn }∞ n=1 is a Cauchy 2 sequence in L (Ft ) equipped with the norm || · || = E| · |2 . If X ∈ L2+ (FT ), then Xn = X and Yn = E[X|Ft ] for n = 1, 2, . . .. Proof. For the two positive integers m and n such that m > n, we have Ym − Yn = E[Xm + m|Ft ] − m − (E[Xn + n|Ft ] − n) = E[Xm + m|Ft ] − E[Xn + m|Ft ] = E[Xn + X1{−m≤X≤−n} + m|Ft ] − E[Xn + m|Ft ].

(29)

Thus, from Lemmas 2.7 and 2.1, we have E(Ym − Yn )2 = E(E[Xn + m|Ft ] − E[Xn + X1{−m≤X≤−n} + m|Ft ])2 ≤ E(E µ [−X1{−m≤X≤−n} |Ft ])2 ≤ eµ

2

(T −t)

E[X 2 1{−m≤X≤−n} ].

(30)

Since X ∈ L2 (FT ), we have lim E[X 2 1{−m≤X≤−n} ] = 0.

m,n→∞

Therefore, lim E(Ym − Yn )2 = 0.

n,m→∞

b Lemma 2.8 shows that E[X|F t ] introduced below is well defined for any X ∈ L (FT ). 2

Definition 2.2. For X ∈ L2 (FT ), denote Xn := X1{X≥−n} .

(31)

We introduce a dynamic operator {Ebs,t [X], X ∈ L2 (FT ); 0 ≤ s ≤ t ≤ T } by and

b n |Ft ] := E[Xn + n|Ft ] − n, E[X

b b E[X|F t ] := lim E[Xn |Ft ] n→∞

for any X ∈ L2 (FT ).

n = 1, 2, . . . ;

(32)

(33)

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34

b Remark 2.5. It is obvious that if X ∈ L2+ (FT ), we have E[X|F t ] = E[X|Ft ] a.s. for t ∈ [0, T ]. b t ]. We have the following continuity of the extended operator E[·|F

Theorem 2.2. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator defined on L2+ (FT ) and satisfy (H1)’ and (H2)’. For each t ∈ [0, T ], the conditional b t ] is strongly continuous from L2+ (FT ) to L2 (Ft ). That expectation operator E[·|F is, if lim Yn = Y, strongly in L2 (FT ),

n→∞

then b n |Ft ] = E[Y b |Ft ], lim E[Y

strongly in L2 (Ft ).

n→∞

(34)

Proof. For the two positive integers m and n such that m > n, define b n |Ft ] − E[Yn 1{Y ≥−m} + m|Ft ] − m δ1 := E[Y n

and

δ2 := (E[Yn 1{Yn ≥−m} + m|Ft ] − m) − (E[Y 1{Y ≥−m} + m|Ft ] − m). Using Lemma 2.7, we have |δ2 | ≤ |E[Y 1{Y ≥−m} + m + |Yn 1{Yn ≥−m} − Y 1{Y ≥−m} ||Ft ] −E[Y 1{Y ≥−m} + m|Ft ]| ≤ E µ [|Yn 1{Yn ≥−m} − Y 1{Y ≥−m} ||Ft ] ≤ E µ [|Yn 1{Yn ≥−m} − Yn ||Ft ] + E µ [|Yn − Y ||Ft ] + E µ [|Y − Y 1{Y ≥−m} ||Ft ]. Thus, using Lemma 2.1, we have E|δ2 |2 ≤ 3EE µ [|Yn 1{Yn ≥−m} − Yn ||Ft ]2 + 3EE µ [|Yn − Y ||Ft ]2 +3EE µ [|Y − Y 1{Y ≥−m} ||Ft ]2 ≤ 3eµ

2

(T −t)

+3eµ ≤ 3eµ

2

2

T

E|Yn 1{Yn ≥−m} − Yn |2

(T −t)

(E|Yn − Y |2 + E|Y − Y 1{Y ≥−m} |2 )

E|Yn 1{Yn St a.s. for t ∈ [r, T ], then we have from (D1) that Et,T [Xn ] ≥ Et,T [X] > St a.s. for t ∈ [r, T ]. In view of (H2), we see that the inequality in (42) is actually an equality for t ∈ [r, T ]. Therefore, e E[X|F t ] = Et,T [X] a.s. for t ∈ [r, T ].

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40

We now prove assertion (ii). Set Xt = Et,T [X] for t ∈ [0, T ]. Then from assertion (i), we have for all 0 ≤ s ≤ t ≤ T e t |Fs ] ≤ Es,t [Xt ] = Xs . E[X

e Therefore, {Et,T [X], 0 ≤ t ≤ T } is an E-supermartingale. Using Lemmas 3.2 and 3.1, we can show the following. Theorem 3.2. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator + with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let Et,T [·] denote the + 2 e restriction of Et,T [·] on L+ (FT ), and E[·|Ft ] be the extension of E [·] given in t,T

Section 2. Then for X ∈ L2 (FT ; ST ), there exists an {Ft , 0 ≤ t ≤ T }-adapted X continuous increasing process {AX t , 0 ≤ t ≤ T } with A0 = 0 such that e + AX |Ft ] − AX for any t ∈ [0, T ]. Et,T [X] = E[X T t

(43)

Proof. Assertion (ii) of Lemma 3.2 shows that {Et,T [X], 0 ≤ t ≤ T } is an e E-supermartingale. In version of (D2) and (H1), we have St ≤ = ≤ ≤

Et,T [X] Et,T [X − ST + ST ] µ Et,T [ST ] + Et,T [X − ST ] µ Et,T [X − ST ].

Therefore, from the first assertion of Lemma 2.1, we have for some positive constant C1 , E sup |Et,T [X]|2 t∈[0,T ]

µ ≤ E sup Et,T [X − ST ]2 + E sup |St |2 t∈[0,T ]

t∈[0,T ]

µ ≤ 2E sup Et,T [|X|]2 + 2E|ST |2 + E sup |St |2 t∈[0,T ]

t∈[0,T ]

µ ≤ 2E sup Et,T [|X|]2 + 3E sup |St |2 t∈[0,T ]

t∈[0,T ]

≤ C1 E[X 2 ] + E sup |St |2 < ∞. t∈[0,T ]

In view of Lemma 3.1, there exists a continuous increasing {Ft , 0 ≤ t ≤ T }X X adapted process {AX t , 0 ≤ t ≤ T } with A0 = 0 such that {Et,T [X]+At , 0 ≤ t ≤ T } e is an E-martingale. Therefore, e T,T [X] + AX |Ft ] = Et,T [X] + AX , E[E T t

∀t ∈ [0, T ].

While ET,T [X] = X, the proof is then complete. In the following, we introduce three lemmas for subsequent arguments. Identical to the case of an {Ft , 0 ≤ t ≤ T }-consistent expectation, we can show

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41

Lemma 3.3. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). For all 0 ≤ s ≤ t ≤ T , X, Y ∈ L2 (Ft ; St ), and B ∈ Fs , we have Es,t [X1B + Y 1B c ] = Es,t [X]1B + Es,t [Y ]1B c .

(44)

Lemma 3.4. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let the process {A X t ,0 ≤ t ≤ T} 2 with AX = 0 be given in Theorem 3.2. For a given X ∈ L (F ; S T T ) and some 0 r ∈ [0, t), if Es,T [X](ω) > Ss (ω),

s ∈ [r, t], a.s. ω ∈ B, B ∈ Fr ,

then X AX s 1B = A t 1B ,

a.s. for s ∈ [r, t].

(45)

Proof. For > 0, set Yt = Et,T [X]1B + 1B c . Then Es,t [Yt ] > Ss , s ∈ [r, t]. From Lemma 3.2, Yt satisfies e t |Fs ]. Es,t [Yt ] = E[Y

(46)

Then from Lemma 3.4, we have for ∀s ∈ [r, t],

e t + (AX − AX )1B |Fs ] = E[ e E[X e + AX − AX |Ft ] + (AX − AX )|Fs ]1B + 1B c E[Y t s T t t s X X e e = E[E[X + AT − As |Ft ]|Fs ]1B + 1B c e + AX − AX |Fs ]1B + 1B c = E[X T

s

= Es,T [X]1B + 1B c , = Ys , a.s.

(47)

and e t |Fs ] = Es,t [Yt ] E[Y

= Es,t [Et,T [X]]1B + 1B c = Ys ,

a.s..

(48)

X Since AX t 1B − As 1B ≥ 0 , we have from (47) and (48) that X AX s 1B = A t 1B ,

a.s. for ∀s ∈ [r, t].

Lemma 3.5. Let {Es,t [·], 0 ≤ s ≤ t ≤ T } be an F-consistent dynamic operator with the negative floor S ≤ 0 and satisfy (H1) and (H2). Let the process {A X t ,0 ≤ 2 t ≤ T } with AX = 0 be given in Theorem 3.2. For a given X ∈ L (F T ; ST ), 0 {Et,T [X], 0 ≤ t ≤ T } satisfies the following Z T (Et,T [X] − St )dAX (49) t = 0, a.s. . 0

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42

Proof. We firstly prove 1{Et,T [X]>St } dAX t = 0 a.s.. Set Bn := {ω : Er,T [X](ω) > Sr (ω), ∀[(t − 1/n) ∨ 0] < r < [(t + 1/n) ∧ T ])}; Cn := ∪{C ∈ F(t−1/n)∨0 : C ⊆ Bn }. Since {Ft , 0 ≤ t ≤ T } is the natural filtration of Bt , augmented by all P -null sets of F, we have ∪∞ n=1 Cn = {Et,T [X] > St }. Lemma 3.4 implies that 1Cn dAX t = 0, a.s. . Therefore, 1{Et,T [X]>St } dAX t =0 For Et,T [X] − St ≥ 0, we have Z T Z (Et,T [X] − St )dAX = t 0

T

a.s. .

(Et,T [X] − St )1{Et,T [X]>St } dAX t = 0.

(50)

0

Theorem 3.3. Theorem 1.1 is true in the case of the negative floor S ≤ 0. Proof. From Theorem 2.3, there exists a function g = g(t, z) : Ω × [0, T ] × Rd satisfying (C2) and (C3) and g(·, ·, 0) ≡ 0, such that the following holds: e |Ft ] = E g [Y |Ft ], ∀Y ∈ L2 (FT ), t ∈ [0, T ]. E[Y

e + AX |Ft ] = E g [X + AX |Ft ] for t ∈ Therefore, for X ∈ L2 (FT ; ST ), we have E[X T T [0, T ]. g X From the definition of E g [X + AX T ] and E [X + AT |Ft ], we know that there is 2 d unique Z ∈ LF (0, T ; R ) such that Z T Z T g X X E [X + AT |Ft ] = X + AT + g(s, Zs )ds − hZs , dBs i, a.s. for any t ∈ [0, T ]. t

Set et := X + X

Z

t

T

g(s, Zs )ds − t

Z

T t

X Zs dBs + AX T − At .

(51)

Then from Lemma 3.2, we have e + AX |Ft ] − AX Et,T [X] = E[X T t

X e (52) = E g [X + AX T |Ft ] − At = Xt . X et = Et,T [X] ≥ St , it follows from Lemma 3.5 that {(X et , Zt , A ), 0 ≤ t ≤ T } Since X t is the solution of RBSDE (X, g, S). That is, et = E r;g,S [X], ∀X ∈ L2 (FT ; ST ). Et,T [X] = X (53) t,T

Remark 3.1. From the proof of Theorem 3.3, {(Et,T [X], Zt , AX t ), 0 ≤ t ≤ T } is the solution of RBSDE (X, g, S). Therefore, the increasing process {AX t ,0 ≤ t ≤ T} is unique for a given X ∈ L2 (FT ; ST ).

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Acknowledgments Both authors thank Professor Shige Peng for his helpful comments. References 1. X. Bao and S. Tang, Representation via Reflected Backward Stochastic Differential Equations of a Filtration-Consistent Nonlinear Dynamic Operator with a Floor, submitted to CRAS. 2. X. Bao and S. Tang, Filtration-consistent dynamic nonlinear evaluation with a general continuous floor and associated reflected backward stochastic differential equations, preprint. 3. F. Coquet, Y. Hu, J. Mémin, and S. Peng, Filtration-consistent nonlinear expectations and related g-expectations, Probability Theory and related Fields, 123, 1–27 (2002). 4. N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7, 1–71 (1997). 5. N. El Karoui, C. Kapoudian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability, 25, 702–737 (1997). 6. N. El Karoui and M. C. Quenez, Programmation dynamique et évalution des actifs contingents en marchés incomplet, C. R. Acad. Sci. Paris, Sér. I, 313, 851–854 (1991). 7. N. El Karoui and M. C. Quenez, Dynamic programming and pricing of contingent claims in incomplete market, SIAM J. Control Optim., 33, 29–66 (1995). 8. N. El Karoui and M. C. Quenez, Nonlinear pricing theory and backward stochastic differential equations, in : Financial Mathematics (ed.: W. J. Runggaldier), Lecture Notes in Mathematics 1656, Springer Verlag, 191-246 (1996). 9. I. Karatzas and S. Shreve, Methods of Mathematical Finance, World Publishing Corporation, Beijing (2004). 10. S. Peng, Backward SDE and related g-expectation, in: N. El Karoui and L. Mazliak (eds.), Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics 364, 141–159 (1997). 11. S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probability Theory and related Fields, 113, 473–499 (1999). 12. S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, Stochastic methods in finance, Lecture Notes in Math., 1856, Springer, Berlin, 165–253 (2004). 13. M. Xu, Contributions to Reflected Backward Stochastic Differential Equations: theory, numerical analysis and simulations, thesis, Shandong University, 57–67 (2005).

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Chap15-YongJiongmin

A LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM FOR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

SHUPING CHEN Department of Mathematics, Zhejiang University, Hangzhou 310027, CHINA E-mail: [email protected] JIONGMIN YONG Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA, and School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA E-mail: [email protected]

A linear quadratic optimal control problem is considered for a stochastic Volterra integral equation. As a necessary condition for the optimality, a forward-backward stochastic Volterra integral equation (FBSVIE, for short) is derived, via a duality principle for stochastic integral equations. Keywords: Linear quadratic optimal control problem, forward-backward stochastic integral equations, duality principle. AMS Mathematics subject classification: 60H10.

1. Introduction Let (Ω, F, lF, P) be a complete filtered probability space, on which a d-dimensional ∆ standard Brownian motion W (·) is defined with lF ={Ft }t≥0 being its natural filtration augmented by all the P-null sets. We consider the following controlled linear (forward) stochastic Volterra integral equation (FSVIE, for short): X(t) = ϕ(t) +

Z th 0

+

i A0 (t, s)X(s) + B0 (t, s)u(s) ds

d Z th X i=1

i

Ai (t, s)X(s) + Bi (t, s)u(s) dWi (s),

0

(1.1) t ∈ [0, T ],

where X(·) is the state and u(·) is the control, taking values in Euclidean spaces lRn and lRm of dimensions n and m, respectively, ϕ(·), Ai (· , ·) and Bi (· , ·) are given 44

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deterministic functions of suitable dimensions. The cost functional associated with (1.1) is given by J(u(·)) = E

Z

T 0

h

i h Q(t)X(t), X(t) i +2 h S(t)X(t), u(t) i + h R(t)u(t), u(t) i dt,

(1.2) where Q(·), S(·), and R(·) are matrix-valued deterministic functions of suitable dimensions. Our optimal control problem is to minimize cost functional (1.2) subject to the state equation (1.1) with the control u(·) taken from a space U of admissible controls. Such a problem is referred to as linear quadratic (LQ, for short) problem for a stochastic Volterra integral equation. LQ problem for ordinary differential equations (ODEs, for short) was studied by Bellman–Glicksberg–Gross in 1958 the first time 5 . Kalman 14 and Letov 16 solved LQ problem (for ODEs) in terms of linear feedback control and Riccati equation in 1960. Since then, many authors extend the theory to various situations. See Refs. 2, 27, and 29 for some summaries of LQ theory for ODEs; see Refs. 20, 18, and 17 for LQ theory of infinite-dimensional (deterministic) systems; see Refs. 28, 8–11, for LQ theory of stochastic differential equations. Some detailed historic remarks on LQ theory can be found in Ref. 34. For some recently relevant works, see Refs. 30, 1, 32, and 31. LQ problem for (deterministic) Volterra integral equations was firstly studied in 1967 by Vinokurov 26 . See also Ref. 22. Since then some extensions were developed (see, for example, Refs. 7, 3, 4, 24, 35, and references cited therein). On the other hand, stochastic Voterra integral equations were studied by several authors 6,25,23 , and some interesting applications were indicated in Ref. 13. In this paper, we are going to study the LQ problem for stochastic Volterra integral equations with the state equation of form (1.1) and cost functional (1.2). Due to the nature of our state equation, one could not use Itˆ o’s formula; all the derivations have to be carried out under integral(s). In our discussion, we will extend/modify some results of backward stochastic Volterra integral equations (BSVIEs, for short) developed in Ref. 33 (see also Ref. 19). By a duality principle for linear stochastic integral equations, we derive a forward-backward stochastic Volterra integral equations (FBSVIEs, for short) whose solvability leads to the existence of critical point of the cost functional. Then under some kind of nonnegativity condition on the cost functional, an optimal control will exist. Note that, in our discussion, we do not a priori impose nonnegativity condition on matrix-valued functions Q(·) and R(·). The case of random coefficients and/or a term like h GX(T ), X(T ) i appears in the cost functional will have much more things involved, and we hope to present the relevant results in a forthcoming paper. The rest of the paper is organized as follows. Some preliminary results concerning the state equation will be presented in Section 2. Section 3 is devoted to a discussion on a minimization problem of a quadratic functional in a Hilbert space which will give us some abstract idea of our LQ problem. In Section 4, we present

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some results on linear BSVIEs, among other things, we mainly modify some results from Ref. 33. In Section 5, we introduce a two-point boundary value problem as a necessary condition for optimal control of our LQ problem. This leads to a linear coupled FBSVIE. 2. State Equation and Optimal Control Problem In this section, we present some preliminaries. To begin with, let us introduce some spaces. First of all, we let lRn×m be the set of all (n × m) matrices, and S n be the set of all (n × n) symmetric matrices. For H = lRm , lRm×d , S n , etc., we define Z T n o L2F (0, T ; H) = ϕ : (0, T ) × Ω → H ϕ(·) is lF-adapted, E |ϕ(t)|2 dt < ∞ , 0 n 2 2 CF ([0, T ]; L (Ω; H)) = ϕ(·) ∈ LF (0, T ; H) t 7→ ϕ(t, ·) is continuous o from [0, T ] to L2 (Ω; lRn ), sup E|ϕ(t)|2 < ∞ , t∈[0,T ] n L2F (Ω; C([0, T ]; H)) = ϕ(·) ∈ L2F (0, T ; H) ϕ(·) has continuous paths, h i o E sup |ϕ(t)|2 < ∞ . t∈[0,T ]

∞ ∞ Spaces L∞ F (0, T ; H), CF ([0, T ]; L (Ω; H)), and LF (Ω; C([0, T ]; H)) can be defined in an obvious way. We point out that any process ϕ(·) ∈ CF ([0, T ]; L2(Ω; H)) is continuous as a map from [0, T ] to L2 (Ω; H), and does not necessarily have continuous paths. Next, for any RBanach space Y, we let L2 (0, T ; Y) be the set of all maps ϕ : T [0, T ] → Y such that 0 |ϕ(t)|2 dt < ∞. The spaces L∞ (0, T ; Y) and C([0, T ]; Y) can be defined in a similar way. Note that we may take Y = L2F (0, T ; H), L∞ (0, T ; H), ∞ 2 2 L∞ F (0, T ; H), LF (Ω; L (0, T ; H)), etc. For example, process Z : [0, T ] × Ω → H 2 2 2 belongs to L (0, T ; LF (0, T ; H)) if it is B([0, T ] ) ⊗ FT -measurable; for almost all t ∈ [0, T ], Z(t, ·) is lF-adapted; and Z TZ T E |Z(t, s)|2 dsdt < ∞. 0

0

We now make the following standing assumptions. (H1) Suppose the following hold:  n×n 2 2  )),   A0 (· , ·) ∈ L (0, T ; L (0, T ; lR   n×n 2 ∞ Ai (· , ·) ∈ L (0, T ; L (0, T ; lR )), 1 ≤ i ≤ d, n×m 2 ∞  B0 (· , ·) ∈ L (0, T ; L (0, T ; lR  )),    n×m 2 ∞ Bi (· , ·) ∈ L (0, T ; L (0, T ; lR )), 1 ≤ i ≤ d.

(2.1)

(H2) Suppose the following hold: Q(·) ∈ L∞ (0, T ; S n ),

S(·) ∈ L∞ (0, T ; lRm×n ),

R(·) ∈ L∞ (0, T ; S m ).

(2.2)

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From (1.1), we see that the value of Ai (t, s) and Bi (t, s) with 0 ≤ t < s ≤ T are irrelevant. Hence, in what follows, we make the following convention: Ai (t, s) = 0,

Bi (t, s) = 0,

0 ≤ t < s ≤ T,

0 ≤ i ≤ d.

(2.3)

Next, we let X = L2F (0, T ; lRn ),

U = L2F (0, T ; lRm ),

and define A : X → X and B : U → X as follows:  Z t d Z t  X    Ai (t, s)X(s)dWi (s), A0 (t, s)X(s)ds +   (AX)(t) = 0

0

i=1

(2.4)

Z t d Z t  X     (Bu)(t) = B (t, s)u(s)ds + Bi (t, s)u(s)dWi (s). 0  0

i=1

0

The following result is concerned with the well-posedness of the state equation (1.1). Proposition 2.1. Let (H1) hold. Then operators A : X → X and B : U → X are bounded and A is quasi-nilpotent, i.e., 1

lim kAk k k = 0.

(2.5)

k→∞

Consequently, (I −A)−1 : X → X is bounded, hence, for any ϕ(·) ∈ X and u(·) ∈ U, state equation (1.1) admits a unique solution X(·) ∈ X . Proof. By (2.4), we have that for any X(·) ∈ X , and t ∈ [0, T ], (note (2.3)) n 2 E (AX)(t) ≤ (d + 1)E ≤ (d + 1)E

Z

t 0

nh Z

d Z t 2 X o Ai (t, s)X(s)|2 ds A0 (t, s)X(s)ds +

T

|A0 (t, s)|2 ds + 0

i=1 d X

esssup |Ai (t, s)|2

i=1 s∈[0,T ]

Thus, by (H1), for some L > 0, Z T Z 2 E (AX)(t) dt ≤ LE 0

0

iZ

T

0

o |X(s)|2 ds . (2.6)

T

|X(s)|2 ds.

(2.7)

0

This implies the boundedness of the operator A. The boundedness of B can be proved similarly. From (2.7), we see that for any k ≥ 1, Z t Z tZ s k 2 k−1 2 2 2 E (A X)(t) ≤ L E (A X)(s) ds ≤ L E (Ak−2 X)(r) drds 0 0 0 Z t k Z t L 2 = L2 (t − s)E (Ak−2 X)(s) ds ≤ · · · ≤ (t − s)k E|X(s)|2 ds, t ∈ [0, T ]. k! 0 0

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Hence, k Z T Z t k (A X)(t)|2 dt ≤ L (t − s)k E|X(s)|2 dsdt k! 0 0 0 Z T Lk T k+1 Lk k+1 2 (T − s) E|X(s)| ds ≤ kX(·)k2X . = (k + 1)! 0 (k + 1)!

k(Ak X)(·)k2X = E

Z

T

This implies that kAk k ≤

Lk T k+1 , (k + 1)!

k ≥ 1.

(2.8)

Thus, A is quasi-nilpotent. Using operators A and B, we can write the state equation (1.1) as X = ϕ + AX + Bu.

(2.9)

Since A is quasi-nilpotent, we have the existence and boundedness of (I − A)−1 . Moreover, one has the expansion: (I − A)−1 =

∞ X

Ak .

(2.10)

k=0

Hence, for any ϕ(·) ∈ X and u(·) ∈ U, state equation (1.1) admits a unique solution X(·) ∈ X given by X = (I − A)−1 (ϕ + Bu) ≡

∞ X

Ak (ϕ + Bu).

(2.11)

k=0

This proves our theorem. By the above result, we see that under (H1)–(H2), the cost functional J(u(·)) is well-defined on U. Consequently, we can state our optimal control problem as follows. Problem (LQ). Minimizing J(u(·)) over u(·) ∈ U. The following notions are concerned with Problem (LQ). Definition 2.1. (i) Problem (LQ) is said to be accessible if inf J(u(·)) > −∞.

u(·)∈U

(2.12)

(ii) Problem (LQ) is said to be (uniquely) solvable if there exists a (unique) u ¯(·) ∈ U such that J(¯ u(·)) = inf J(u(·)). u(·)∈U

(2.13)

Any control u ¯(·) ∈ U satisfying (2.13) is called an optimal control, and the cor¯ responding state process X(·) is called an optimal state process. We also call ¯ (X(·), u ¯(·)) an optimal pair.

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Our goals are: (a) Characterize the accessibility and solvability of Problem (LQ), and (b) in the case that Problem (LQ) is solvable, find or characterize optimal controls. 3. Quadratic Functionals in Hilbert Spaces In this section, we briefly look at some properties of quadratic functionals in Hilbert spaces which will be helpful for us to understand Problem (LQ) from a functional analysis point of view. Let H be a Hilbert space and Φ : H → H be a bounded self-adjoint operator. We denote R(Φ) and N (Φ) to be the range and kernel of Φ, respectively. Since Φ is

self-adjoint, N (Φ)⊥ = R(Φ) (and we always have Φ R(Φ) ⊆ R(Φ)). Thus, under

the decomposition H = N (Φ) ⊕ R(Φ), we have the following representation for Φ: ! 00 Φ= (3.1) b , 0Φ

b : R(Φ) → R(Φ) is self-adjoint. Now, we define the pseudo-inverse Φ† by where Φ the following: ! 0 0 † Φ = (3.2) b −1 , 0Φ with domain

D(Φ† ) = N (Φ) + R(Φ) ≡ {u0 + u1 u0 ∈ N (Φ), u1 ∈ R(Φ)} ⊇ R(Φ).

(3.3)

From the above, we can easily seen the following facts:

(i) Φ† is (closed, densely defined, and) self-adjoint; R(Φ) is closed if and only if Φ is bounded. (ii) By the definition of Φ† (see (3.2)), together with (3.3), one has that †

ΦΦ† Φ = Φ,

Φ† ΦΦ† = Φ† ,

(Φ† )† = Φ.

(3.4)

Thus, by (i), R(Φ† ) is closed since Φ is bounded. (iii) Although D(Φ† ) is not necessarily closed, the operator ΦΦ† : D(Φ† ) → H is an orthogonal projection onto R(Φ). Thus, we may naturally extend it, still denoted it by itself, to D(Φ† ) = H. Hence, ΦΦ† : H → R(Φ) ⊆ H is the orthogonal projection onto R(Φ). Note that since Φ is bounded, Φ† Φ is an orthogonal projection from H onto R(Φ† ) = N (Φ† )⊥ = N (Φ)⊥ = R(Φ). Therefore, in fact, we have ΦΦ† = Φ† Φ ≡ PR(Φ) ≡ orthogonal projection onto R(Φ).

(3.5)

(iv) The map Φ 7→ Φ† is not continuous (which can be seen even from onedimensional case).

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Now, let us consider a quadratic functional on H: J(u) = h Φu, u i +2 h v, u i,

u ∈ D(Φ) ⊆ H,

(3.6)

where Φ : D(Φ) ⊆ H → H is a self-adjoint linear operator and v ∈ H. The following result is concerned with the completing square and critical point(s) of the functional J(·). Note here that we do not assume positive (negative) semi-definite condition on Φ. Proposition 3.1. (i) There exists a u ˆ ∈ H such that J(u) = h Φ(u − u ˆ), u − u ˆ i − h Φˆ u, u ˆ i,

∀u ∈ H,

(3.7)

if and only if v ∈ R(Φ) ⊆ D(Φ† ) .

(3.8)

(ii) Any u ˆ ∈ H satisfies (3.7) if and only if it is a solution of the following equation: Φˆ u + v = 0,

(3.9)

which is equivalent to the following: u ˆ = −Φ† v + (I − Φ† Φ)e v,

(3.10)

for some ve ∈ H (in particular, u ˆ = −Φ† v is a solution). (iii) When (3.7) holds, it is necessary that J(u) = h Φ(u − u ˆ), u − u ˆ i − h Φ† v, v i,

∀u ∈ H.

(3.11)

Moreover, u ˆ is unique if and only if N (Φ) = {0}. Proof. (i) For any u ˆ ∈ H, on has J(u) ≡ h Φu, u i +2 h v, u i = h Φ(u − u ˆ), u − u ˆ i +2 h Φˆ u + v, u i − h Φˆ u, u ˆ i,

∀u ∈ H.

(3.12)

Hence, there exists a u ˆ ∈ H such that (3.7) holds if and only if (3.9) holds, which gives (3.8) (and the first part of (ii)). Conversely, if (3.8) holds, then there exists a u ˆ ∈ H such that (3.9) holds. Consequently, h Φ(u − u ˆ), u − u ˆ i − h Φˆ u, u ˆ i = h Φu, u i −2 h Φˆ u, u i + h Φˆ u, u ˆ i − h Φˆ u, u î = h Φu, u i +2 h v, u i = J(u),

(3.13)

proving (3.7). (ii) We have proved the first part of (ii) (from (3.12)). The second part is straightforward.

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(iii) For any u ˆ ∈ H satisfying (3.7), one must have (3.9). Hence, h Φ(u − u ˆ), u − u ˆ i − h Φ† v, v i = h Φu, u i −2 h Φˆ u, u i + h Φˆ u, u ˆ i − h Φ† Φˆ u, Φˆ ui = h Φu, u i +2 h v, u i = J(u), (3.14) which proves (3.11). Finally, by (3.9), we see that u ˆ is unique if and only if N (Φ) = {0}. Note that (3.9) is equivalent to the following: 0 = Φˆ u+v ≡

1 ∇J(ˆ u). 2

(3.15)

Thus, u ˆ is actually a critical point of functional J(·). Hence, Proposition 3.1 characterizes critical points of the quadratic functional J(·). Equations (3.7) and (3.11) are completion of square for the functional J(·) (although Φ is not necessarily positive/negative semi-definite). Next, for any self-adjoint operator Φ, regardless whether it is bounded or unbounded, we have the following spectrum decomposition 12 Z Φ= λdPλ , (3.16) σ(Φ)

where σ(Φ) ⊆ lR is the spectrum of Φ, (which is a compact set if Φ is bounded, and it is unbounded if Φ is unbounded); and {Pλ λ ∈ σ(Φ)} is a family of projection measures. In the case that Φ ≥ 0,

(3.17)

one has from (3.16) that σ(Φ) ⊆ [0, ∞), and  Z    Φα = λα dPλ , ∀α ≥ 0,  σ(Φ) Z   † α α †  λ−α dPλ ,  (Φ ) = (Φ ) =

(3.18) ∀α > 0.

σ(Φ)\{0}

Now, we can consider minimization problem for functional J(·). Proposition 3.2. Let Φ : H → H be bonded and self-adjoint, and v ∈ H. (i) The following holds: inf J(u) > −∞,

u∈H

(3.19)

if and only if (3.17) holds and 1

v ∈ R(Φ 2 ).

(3.20)

In this case, 1

inf J(u) = −|(Φ† ) 2 v|2 .

u∈H

(3.21)

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(ii) There exists a u ˆ ∈ H such that J(ˆ u) = inf J(u),

(3.22)

u∈H

if and only if (3.17) and (3.8) hold; and in this case, all the conclusions in Proposition 3.1 hold. Proof. We need only to prove (i). First, let (3.19) hold. It is straightforward that one must have (3.17). Next, we prove (3.20) by contradiction. Suppose (3.20) does not hold. For any n ≥ 1, let Z dPλ v. vn = 1 ,n] σ(Φ)∩[ n

Then vn ∈ R(Φ), and Z † h v, Φ vn i =

1

λd|Pλ v|2 = |(Φ† ) 2 vn |2 → ∞,

1 σ(Φ)∩[ n ,n]

n → ∞.

Hence, letting un = −Φ† vn , we obtain 1

J(un ) = h Φun , un i +2 h v, un i = −|(Φ† ) 2 vn |2 → −∞,

n → ∞,

contradicting (3.19). Conversely, if (3.17) and (3.20) hold, then for any u ∈ H, one has 1

1

1

1

1

1

1

J(u) = |Φ 2 u|2 +2 h(Φ† ) 2 v, Φ 2 u i = |Φ 2 u+(Φ† ) 2 v|2 −|(Φ† ) 2 v|2 ≥ −|(Φ† ) 2 v|2 > −∞. (3.23) Hence, sufficiency follows. Finally, from the fact that 1

1

R((Φ† ) 2 ) ⊆ R(Φ 2 ) = R(Φ), we can always find a sequence un ∈ H so that (note (3.23)) 1

1

1

1

J(un ) = |Φ 2 un + (Φ† ) 2 v|2 − |(Φ† ) 2 v|2 → −|(Φ† ) 2 v|2 ,

n → ∞.

Thus, (3.21) follows. The above result tells us that the existence of minimum is strictly stronger than the finiteness of the infimum of the functional J(·), which have been described by 1 conditions (3.8) and (3.20), respectively. Note here that R(Φ) ⊆ R(Φ 2 ) when (3.17) holds. The following example shows the necessity of condition (3.20) in a concrete way. Example 3.1. Let H = `2 . For any u = {ai }∞ i=1 ∈ H, define Φu by Φu = {β i−1 ai }∞ i=1 ,

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where β ∈ (0, 1). Then Φ : H → H is bounded, self-adjoint, and positive definite (but not uniformly). Let v = {i−1 }∞ lim Φun i=1 ∈ H1 . Then v ∈ R(Φ) since v = n→∞ with 1 1 1 , ,··· , , 0, 0, · · · } ∈ H. un = {1, 2β 3β 2 nβ n−1 But, clearly, v ∈ / R(Φ). Now, consider a quadratic functional ∞ X 2ai J(u) = h Φu, u i +2 h v, u i = β i−1 a2i + . i i=1 Then by letting un as above, we see that J(−un ) = h Φun , un i −2 h v, un i n h n X X 2 i 1 1 = β i−1 2 2(i−1) − 2 i−1 = − → −∞, 2 β (i−1) i β i β i i=1 i=1

as n → ∞.

This means that inf J(u) = −∞.

u∈H

An interesting point here is that positive semi-definiteness of Φ does not even ensure the finiteness of the infimum of J(·). Now, we return to Problem (LQ). Denote ( QX = Q(·)X(·), SX = S(·)X(·), Ru = R(·)u(·),

∀u(·) ∈ U.

∀X(·) ∈ X ,

(3.24)

Then under (H1)–(H2) and with ϕ(·) ∈ L2F (0, T ; lRn ), the cost functional (1.2) can be written as follows: J(u) = h QX, X i +2 h SX, u i + h Ru, u i Q S∗ (I − A)−1 (ϕ + Bu) (I − A)−1 (ϕ + Bu) =h , i S R u u Q S∗ (I − A)−1 (I − A)−1 B ϕ (I − A)−1 (I − A)−1 B ϕ =h , i S R 0 I u 0 I u = h Φ2 u, u i +2 h Φ1 ϕ, u i + h Φ0 ϕ, ϕ i, (3.25) where   Φ = (I − A∗ )−1 Q(I − A)−1 ,   0 n o Φ1 = B ∗ (I − A∗ )−1 Q + S (I − A)−1 ,    Φ2 = B ∗ (I − A∗ )−1 Q(I − A)−1 B + S(I − A)−1 B + B ∗ (I − A∗ )−1 S ∗ + R. (3.26) Consequently, by Propositions 3.1 and 3.2, we obtain the following abstract result.

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Proposition 3.3. (i) If Problem (LQ) admits an optimal control, then Φ2 ≥ 0.

(3.27)

(ii) When (3.27) holds, Problem (LQ) admits an optimal control if and only if Φ1 ϕ ∈ R(Φ2 ).

(3.28)

In this case, u ¯ is an optimal control if and only if it is a solution of the equation: Φ2 u ¯ + Φ1 ϕ = 0,

(3.29)

u ¯ = −Φ†2 Φ1 ϕ + (I − Φ†2 Φ2 )v,

(3.30)

which is given by

for some v ∈ U, with the optimal cost functional J(¯ u(·)) = h(Φ0 − Φ∗1 Φ†2 Φ1 )ϕ, ϕ i .

(3.31)

Further, if Φ2 is invertible, the optimal control is unique. Although the above gives necessary and sufficient conditions under which Problem (LQ) is (uniquely) solvable, the conditions imposed a little too abstract, for example, it is by no means obvious how one can represent A∗ and B ∗ . Therefore, we need to make further efforts in our investigation.

4. BSVIEs and Duality Principle In this section, we recall/modify some results on BSVIEs from Ref. 33, which will be useful in studying Problem (LQ). To begin with, let us consider the following BSVIE introduced in Ref. 33: Z T Z T Y (t) = ψ(t) + g(t, s, Y (s), Z(s, t))ds − Z(t, s)dW (s), t ∈ [0, T ], (4.1) t

t

n

n×d

where g : [0, T ]2 × lR × lR the following definition.

n

→ lR is a given map. According to Ref. 33, we have

Definition 4.1. A pair of process (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) is called an adapted solution of (4.1) if for almost all t ∈ [0, T ], almost surely, (4.1) is satisfied in the Itˆ o sense. In Ref. 33, it was proved that under the condition that (y, z) 7→ g(t, s, y, z) is uniformly Lipschitz, there exists a unique adapted solution (Y (·), Z(· , ·)) to (4.1). Based on this, some further results were established (see Ref. 33).

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Before going further, let us look at a simple situation. For any Y (·) ∈ L2F (0, T ; lRn ) and Z(· , ·) ∈ L2 (0, T ; L2F (0, T ; lRn )), we call Z T ψ(t) = Y (t) + Z(t, s)dW (s), t ∈ [0, T ]. (4.2) t

2

Then ψ(·) ∈ L ((0, T ) × Ω), and ψ(·) is not necessary lF-adapted. (Y (·), Z(· , ·)) is an adapted solution of the following BSVIE: Z T Y (t) = ψ(t) − Z(t, s)dW (s), t ∈ [0, T ].

Clearly,

(4.3)

t

On the other hand, for ψ(·), by martingale representation theorem, there exists a b , ·) ∈ L2 (0, T ; L2 (0, T ; lRn )) such that unique Z(· F Z T b s)dW (s), ψ(t) = Eψ(t) + Z(t, t ∈ [0, T ]. (4.4) 0

If we define

∆ Yb (t) = Eψ(t) +

Z

t 0

b s)dW (s), Z(t,

t ∈ [0, T ],

b , ·)) is also an adapted solution of (5.14). Hence, we have then (Yb (·), Z(· Z Th i b s) − Z(t, s) dW (s), b Z(t, t ∈ [0, T ]. Y (t) − Y (t) =

(4.5)

(4.6)

t

By taking conditional expectation, we see that (note (4.5) and Eψ(t) = EY (t)) Z t b s)dW (s), b Z(t, t ∈ [0, T ]. (4.7) Y (t) = Y (t) = EY (t) + 0

Hence, we must have

b s), Z(t, s) = Z(t,

s ∈ [t, T ].

(4.8)

In such a sense, the adapted solution to (5.14) is unique. But, in general, (4.8) might not be true for s ∈ [0, t]. Hence, we might not have (comparing with (4.5) and (4.7)) Z t Y (t) = EY (t) + Z(t, s)dW (s), t ∈ [0, T ]. (4.9) 0

To further convince ourselves, let us look at the following example. Example 4.1. Example 4.2. Take d = 1, Y (t) = W (t), and Z(t, s) ≡ 2. Then Z T ψ(t) = Y (t) + Z(t, s)dW (s) = 2W (T ) − W (t), t ∈ [0, T ], t

b s) = 2 − and (trivially) (5.14) holds. On the other hand, (4.5) holds with Z(t, I[0,t] (s). Clearly, (4.8) holds, but b s), Z(t, s) 6= Z(t,

s ∈ [0, t],

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and Y (t) = W (t) 6= 2W (T ) − W (t) = EY (t) +

Z

T

Z(t, s)dW (s),

t ∈ [0, T ]. (4.10)

t

The above shows the following: (i) The uniqueness of adapted solution (Y (·), Z(· , ·)) to BSVIE (4.1) does not give the uniqueness of Z(t, s) for s ∈ [0, t], and (ii) in general, we do not have (4.9) for any adapted solution (Y (·), Z(· , ·)). However, from Ref. 33, we know that relation (4.9) played an important role in establishing duality relation. Hence, the following notion will be useful. Definition 4.2. A pair of process (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) is called an adapted M-solution of (4.1) if (4.1) is satisfied in the Itˆ o sense, and (4.9) holds. In the above, “M” in “M-solution” stands for “Martingale Representation”. Now, we state the following result concerning the existence and uniqueness of adapted M-solutions of BSVIE (4.1). Proposition 4.1. Suppose that (y, z) 7→ g(t, s, y, z) is uniformly Lipschitz continuous. Then for any ψ(·) ∈ L2 ((0, T ) × Ω), (4.1) admits a unique adapted M-solution (Y (·), Z(· , ·)). Proof. Let M be the set of all processes (y(·), z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )) satisfying Z t y(t) = Ey(t) + z(t, s)dW (s), t ∈ [0, T ]. 0

Then M is a (nontrivial) closed subspace of L2F (0, T ; lRn )×L2 (0, T ; L2F (0, T ; lRn×d )). Now, for any (y(·), z(· , ·)) ∈ M, let Z T ˆ = ψ(t) + ψ(t) g(t, s, y(s), z(s, t))ds, t ∈ [0, T ]. t

By martingale representation theorem, there exists a unique Z(· , ·) L2 (0, T ; L2F (0, T ; lRn×d )) such that Z T ˆ = E ψ(t) ˆ + ψ(t) Z(t, s)dW (s), t ∈ [0, T ]. 0

Next, we define ˆ + Y (t) = E ψ(t)

Z

t

Z(t, s)dW (s),

Then (Y (·), Z(· , ·)) ∈ M and Z T Z Y (t) = ψ(t) + g(t, s, y(s), z(s, t))ds − t

t ∈ [0, T ].

0

T

Z(t, s)dW (s), t

t ∈ [0, T ].

∈

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Hence, we obtain a map Φ : M → M, (y(·), z(· , ·)) 7→ (Y (·), Z(· , ·)). Then by a contraction mapping argument similar to that in Ref. 33, we can show that Φ admits a unique fixed point, which is the adapted M-solution that we are looking for. Note that unlike adapted solution, the uniqueness of adapted M-solution really gives the uniqueness of Z(· , ·) over [0, T ] × [0, T ], due to the additional requirement (4.9). In Example 4.1, we see that although (Y (·), Z(· , ·)) is merely an adapted solution b , ·)) is the adapted M-solution and (4.9) fails, we can modify Z(· , ·) so that (Y (·), Z(· (therefore (4.7) holds). Sometime, we might need some other restriction other than (4.9). Inspired by the above, we have the following result. Proposition 4.2. Let (y, z) 7→ g(t, s, y, z) be uniformly Lipschitz continuous. Then for any ψ(·) ∈ L2 ((0, T )×Ω) and ϕ(·) ∈ L2F (0, T ; lRn ), (4.1) admits a unique adapted solution (Y (·), Z(· , ·)) satisfying Z t Z(t, s)dW (s), t ∈ [0, T ]. (4.11) ϕ(t) = Eϕ(t) + 0

Proof. First of all, by martingale representation theorem, we can find ζ(· , ·) ∈ L2 (0, T ; L2F (0, T ; lRn )) such that Z t ϕ(t) = Eϕ(t) + ζ(t, s)dW (s), t ∈ [0, T ]. (4.12) 0

Note that since ϕ(·) is lF-adapted, ζ(t, s) = 0 for s ∈ [t, T ]. Next, consider the following BSVIE: Z T Z T Y (t) = ψ(t) + g(t, s, Y (s), ζ(s, t))ds − Z(t, s)dW (s), t ∈ [0, T ]. (4.13) t

t

By Proposition 4.1, the above admits a unique adapted solution (Y (·), Z(· , ·)). Note that in (4.13), only the values Z(t, s) with 0 ≤ s ≤ t ≤ T are used and by changing the values Z(t, s) for 0 ≤ s ≤ t ≤ T only, (4.13) remains unchanged. On the other hand, in the drift term, only ζ(t, s) with 0 ≤ s ≤ t ≤ T are used. Hence, it we redefine Z(t, s) = ζ(t, s),

0 ≤ s ≤ t ≤ T,

then (Y (·), Z(· , ·)) is still an adapted solution of (4.1) and (4.11) holds. Note of course that the adapted solution (Y (·), Z(· , ·)) obtained in Proposition 4.2 is not an adapted M-solution. Actually, (4.9) is replaced by (4.11). Also, we see that in the proof of Proposition 4.2, we have taken the advantage that the drift does not depending on Z(t, s) (only depending on Z(s, t)). In the case that the drift also depends on Z(t, s), we will still have a unique adapted M-solution. But, it is not clear if Proposition 4.2 holds.

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Now, we strengthen the standing assumptions (H1) for the coefficients of the state equation as follows. (H3) Suppose the following hold: Ai (· , ·) ∈ L∞ ((0, T )2 ; lRn×n ),

Bi (· , ·) ∈ L∞ ((0, T )2 ; lRn×m ),

0 ≤ i ≤ d. (4.14)

The following result gives representations for A∗ and B ∗ in terms of BSVIEs. Theorem 4.1. Suppose (H3) holds. Let A and B be defined by (2.4). Then for any ψ(·) ∈ L2F (0, T ; lRn ) and v(·) ∈ L2F (0, T ; lRm ), (A∗ ψ)(t) = η(t),

t ∈ [0, T ], a.s. ,

(4.15)

(B ∗ v)(t) = λ(t),

t ∈ [0, T ], a.s. ,

(4.16)

and

where (η(·), ζ(· , ·)) is the unique adapted M-solution to the following BSVIE:

η(t) =

Z

T t

h

A0 (s, t)T ψ(s)+

d X i=1

Z i Ai (s, t)T ζi (s, t) ds−

T

ζ(t, s)dW (s),

t ∈ [0, T ],

t

(4.17)

satisfying ψ(t) = Eψ(t) +

Z

t

ζ(t, s)dW (s),

t ∈ [0, T ],

(4.18)

0

and (λ(·), µ(· , ·)) is the unique adapted M-solution to the following BSVIE:

λ(t) =

Z

T t

h

T

B0 (s, t) v(s)+

d X

T

i

Bi (s, t) µi (s, t) ds−

i=1

Z

T

µ(t, s)dW (s),

t ∈ [0, T ],

t

(4.19)

satisfying v(t) = Ev(t) +

Z

t

µ(t, s)dW (s),

t ∈ [0, T ].

(4.20)

0

Proof. By Proposition 4.2, BSVIE (4.17) admits a unique adapted solution (η(·), ζ(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn )) satisfying (4.18). Conse-

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quently, for any X(·) ∈ L2F (0, T ; lRn ), one has E

Z

T

h(A∗ ψ)(t), X(t) i dt ≡ E 0

=E

Z

=E

Z

T 0 T 0

+E =E

Z

T 0

=E

Z

=E

Z

n Z

Z

h 0

T s

h ψ(t), (AX)(t) i dt 0

A0 (t, s)X(s)ds + 0

d Z X i=1

0

Z

0

d Z X

t

ζ(t, s)dW (s), 0

i=1

h A0 (s, t)T ψ(s), X(t) i dsdt +

t

Z

t

o Ai (t, s)X(s)dWi (s) i dt

h A0 (t, s)T ψ(t), X(s) i dtds h Eψ(t) +

T

T

t

T

Z

T

Z

h ψ(t),

Z

t

Ai (t, s)X(s)dWi (s) i dt 0

d X

E

i=1

T t

h

A0 (s, t)T ψ(s) +

T

h η(t) + 0

Z

d X i=1

Z

T 0

Z

t

h ζi (t, s), Ai (t, s)X(s) i dsdt 0

i Ai (s, t)T ζi (s, t) ds, X(t) i dt

T

ζ(t, s)dW (s), X(t) i dt = E t

Z

T

h η(t), X(t) i dt. 0

Since X(·) ∈ L2F (0, T ; lRn ) is arbitrary, and η(·) ∈ L2F (0, T ; lRn ), we obtain (4.15). Representation (4.16) can be proved similarly. Let us now consider the following FSVIE: X(t) = f (t) +

Z

t

A0 (t, s)X(s)ds + 0

d Z X i=1

t

Ai (t, s)X(s)dWi (s),

t ∈ [0, T ], (4.21)

0

with f (·) ∈ L2F (0, T ; lRn ). It is clear that our state equation (1.1) is a special case of the above with Z t d Z t X f (t) = ϕ(t) + B0 (t, s)u(s)ds + Bi (t, s)u(s)dWi (s), t ∈ [0, T ]. (4.22) 0

i=1

0

The following result is called a duality principle. Theorem 4.2. Let Ai (· , ·) (0 ≤ i ≤ d) satisfy (H3) and f (·) ∈ L2F (0, T ; lRn ), g(·) ∈ L2 ((0, T ) × Ω; lRn ). Let X(·) ∈ L2F (0, T ; lRn ) be the solution of FSVIE (4.21), and (Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn )×L2 (0, T ; L2F (0, T ; lRn×d )) be the adapted solution to the following BSVIE: Y (t) = g(t) +

Z

T t

h

A0 (s, t)T Y (s) +

d X i=1

−

i Ai (s, t)T Zi (s, t) ds Z

t

(4.23)

T

Z(t, s)dW (s),

t ∈ [0, T ],

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satisfying Y (t) = EY (t) +

Z

t

Z(t, s)dW (s),

t ∈ [0, T ].

(4.24)

0

Then the following relation holds: Z T Z E h X(t), g(t) i dt = E 0

T

h f (t), Y (t) i dt.

(4.25)

0

We call (4.23)–(4.24) the adjoint equation of FSVIE (4.21), and call (4.25) a duality relation between (4.21) and (4.23)–(4.24). We point out that process g(·) appeared in the above is not required to be lF-adapted. The above result modifies that in Ref. 33. Proof. By Theorem 4.1, we see that (4.23) with (4.24) can be written as Y = g + A∗ Y. Since (I − A)−1 exists and bounded, we have Y = (I − A∗ )−1 g. Hence, h X, g i = h(I − A)−1 f, g i = h f, (I − A∗ )−1 g i = h f, Y i, proving (4.25). 5. A Maximum Principle and FBSVIEs In this section, we would like to seek some alternative conditions for the solvability of Problem (LQ). We begin with a necessary condition for optimality. Suppose ¯ (X(·), u ¯(·)) is an optimal pair of Problem (LQ). Then, by a standard variational technique, we have the following: For any u(·) ∈ U, nZ T h i o ¯ + S(t)T u ¯ + R(t)¯ E h Q(t)X(t) ¯(t), X(t) i + h S(t)X(t) u(t), u(t) i dt = 0, 0

(5.1)

where X(t) =

Z th 0

d Z th i i X A0 (t, s)X(s) + B0 (t, s)u(s) ds + Ai (t, s)X(s) + Bi (t, s)u(s) dWi (s)

≡ f (t) +

Z

t

A0 (t, s)X(s)ds + 0

d Z t X

i=1

0

Ai (t, s)X(s)dWi (s),

t ∈ [0, T ],

0

i=1

(5.2)

with f (t) =

Z

t

B0 (t, s)u(s)ds + 0

d Z X i=1

t

Bi (t, s)u(s)dWi (s), 0

t ∈ [0, T ].

(5.3)

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By the duality principle established in the previous section, we have the following result which can be regarded as a Pontryagin’s maximum principle. ¯ Theorem 5.1. Let (H2)-(H3) hold. Let (X(·), u ¯(·)) be an optimal pair of Problem (LQ). Then there exists adapted solutions (Y (·), Z(· , ·)), (λ(·), µ(· , ·)) to the following BSVIEs: Z

¯ + S(t)T u Y (t) = Q(t)X(t) ¯(t) + −

Z

T t

h

T

A0 (s, t) Y (s) +

d X i=1

i Ai (s, t)T Zi (s, t) ds

T

Z(t, s)dW (s), t

(5.4)

and λ(t) =

Z

h

T t

B0 (s, t)T Y (s) +

d X i=1

Z i Bi (s, t)T Zi (s, t) ds −

T

µ(t, s)dW (s),

(5.5)

t

such that ¯ + R(t)¯ S(t)X(t) u(t) + λ(t) = 0,

a.e. t ∈ [0, T ], a.s.

(5.6)

¯ We see that (1.1) (with (X(·), u(·)) replaced by (X(·), u ¯(·))) together with (5.4)– (5.5) is a system of coupled forward and backward stochastic Volterra integral equations. The coupling is given through (5.6). We call such a system an FBSVIE. By Proposition 4.1, BSVIEs (5.4) and (5.5) admit unique adapted

Proof. solutions

(Y (·), Z(· , ·)) ∈ L2F (0, T ; lRn ) × L2 (0, T ; L2F (0, T ; lRn×d )), (λ(·), µ(· , ·)) ∈ L2F (0, T ; lRm ) × L2 (0, T ; L2F (0, T ; lRm×d )), ¯ + S(·)T u respectively. By Theorem 4.1, with g(·) = Q(·)X(·) ¯(·), and f (·) given by (4.22), we have

E

Z

T 0

=E

Z

¯ + S(t)T u h X(t), Q(t)X(t) ¯(t) i dt T

h 0

=E

Z

=E

Z

T 0

t

B0 (t, s)u(s)ds + 0

Z th

d Z X i=1

h

Z

T t

t

Bi (t, s)u(s)dWi (s), Y (t) i dt 0

h B0 (t, s)T Y (t), u(s) i + h

0

T 0

Z

d X i=1

h

B0 (s, t)T Y (s) +

d X i=1

i Bi (t, s)T Zi (t, s), u(s) i dsdt

i Bi (s, t)T Zi (s, t) ds, u(t) i dt.

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Hence, (5.1) implies Z T ¯ + R(t)¯ 0=E h S(t)X(t) u(t) 0

+ =E =E

Z

Z

T t

h

T

h λ(t) + 0

Z

T

B0 (s, t)T Y (s) + Z

d X i=1

T

i Bi (s, t)T Zi (s, t) ds, u(t) i dt

(5.7)

¯ + R(t)¯ µ(t, s)dW (s) + S(t)X(t) u(t), u(t) i dt

t

¯ + R(t)¯ h λ(t) + S(t)X(t) u(t), u(t) i dt,

∀u(·) ∈ U[0, T ].

0

Consequently, (5.6) follows. Let us look at the above in a little point view, using the results from Section 3. To this end, let us denote X u (·) and X ϕ (·) to be the solution of (1.1) corresponding to (u(·), 0) and (0, ϕ(·)), respectively. Then X(·) = X u (·) + X ϕ (·). Now, by the definition of Φ2 , we have that for any u(·) ∈ U, (Φ2 u)(t) = B ∗ (I − A∗ )−1 [QX u + S T u] + SX u + Ru (t) = (B ∗ Y u )(t) + S(t)X u (t) + R(t)u(t) = λu (t) + S(t)X u (t) + R(t)u(t), (5.8)

with  Z Th d  i X   u u T T u   Y (t) = Q(t)X (t) + S(t) u(t) + A0 (s, t) Y (s) + Ai (s, t)T Ziu (s, t) ds    t  i=1   Z T u − Z (t, s)dW (s),   t    Z Th Z T d  i X   u T u T u  λ (t) = B (s, t) Y (s) + B (s, t) µ (s, t) ds − µu (t, s)dW (s),  0 i i  t

i=1

t

(5.9)

satisfying

 Z t   u u  Y (t) = EY (t) + Z u (t, s)dW (s),  Z 0t   u u   λ (t) = Eλ (t) + µu (t, s)dW (s),

t ∈ [0, T ].

(5.10)

0

Thus, Φ2 ≥ 0 is equivalent to the following: Z T E h λu (t) + S(t)X u (t) + R(t)u(t), u(t) i dt ≥ 0,

∀u(·) ∈ U,

(5.11)

0

On the other hand,

(Φ1 ϕ)(t) = B ∗ (I − A∗ )−1 QX ϕ (t) + S(t)T X ϕ (t).

(5.12)

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Hence, (Φ2 u + Φ1 ϕ)(t) = B ∗ (I − A∗ )−1 [QX + S T u] + SX + Ru (t)

(5.13)

= (B ∗ Y )(t) + S(t)X(t) + R(t)u(t) = λ(t) + S(t)X(t) + R(t)u(t),

with  Z Th d  i X   T T T   Y (t) = Q(t)X(t) + S(t) u(t) + A (s, t) Y (s) + A (s, t) Z (s, t) ds 0 i i    t  i=1   Z T − Z(t, s)dW (s),   t    Z Th Z T d  i X   T T  B0 (s, t) Y (s) + Bi (s, t) µi (s, t) ds − µ(t, s)dW (s),   λ(t) = t

i=1

t

(5.14)

satisfying

 Z t    Z(t, s)dW (s),  Y (t) = EY (t) + 0 Z t     λ(t) = Eλ(t) + µ(t, s)dW (s),

t ∈ [0, T ].

(5.15)

0

Note that the equation for (λ(·), µ(· , ·)) in (5.14) is different from the equation (5.5). But, as we explained in the previous section, we can redefine µ(· , ·) in (5.5) so that it takes the same form as that in (5.14). From the above, we obtain the following. Theorem 5.2. Problem (LQ) is solvable if and only if (5.11) holds and for the given ϕ(·) ∈ L2F (0, T ; lRn ), there exists a u(·) ∈ U ≡ L2F (0, T ; lRm ) such that λ(t) + S(t)X(t) + R(t)u(t) = 0,

t ∈ [0, T ],

(5.16)

with λ(·) given through (5.14)–(5.15). In this case, (X(·), u(·)) is an optimal pair. The above gives alternative conditions for the solvability of Problem (LQ). In some sense, conditions in Theorem 5.2 are a little less abstract than those presented in Section 3. Note that if R(t)−1 exists and uniformly bounded. Then (5.16) is equivalent to h i u(t) = −R(t)−1 S(t)X(t) + λ(t) ,

t ∈ [0, T ].

(5.17)

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Substituting the above into (1.1) and (5.14), we obtain the following:  Z tn o    X(t) = ϕ(t) + [A0 (t, s) − B0 (t, s)R(s)−1 S(s)]X(s) − B0 (t, s)R(s)−1 λ(s) ds    0    d Z tn  o X   −1 −1  [A (t, s) − B (t, s)R(s) S(s)]X(s) − B (t, s)R(s) λ(s) dWj (s), +  j j j    j=1 0   Y (t) = [Q(t) − S(t)T R(t)−1 S(t)]X(t) − S(t)T R(t)−1 λ(t)   Z Th Z T d  i X   T T  + A0 (s, t) Y (s) + Ai (s, t) Zi (s, t) ds − Z(t, s)dW (s),    t t  i=1    Z Th Z T  d i  X  T T   λ(t) = B (s, t) Y (s) + B (s, t) µ (s, t) ds − µ(t, s)dW (s), 0 i i  t

t

i=1

(5.18)

with constraints:

 Z t    Z(t, s)dW (s),  Y (t) = EY (t) + 0 Z t     λ(t) = Eλ(t) + µ(t, s)dW (s).

(5.19)

0

This is a coupled FBSVIE. The general solvability problem for the above equation is still under our careful investigation. We hope to report some further results related to this in the near future.

References 1. M. Ait Rami, J. B. Moore, and X. Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim. 40, 1296–1311 (2001/02). 2. B. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. 3. S. A. Belbas, Iterative schemes for optimal control of Volterra integral equations, Nonlinear Anal. 37, 57–79 (1999). 4. S. A. Belbas and W. H. Schmidt, Optimal control of Volterra equations with impulses, Appl. Math. Computation 166, 696–723 (2005). 5. R. Bellman, I. Glicksberg, and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, Rand Corporation, Santa Monica, CA, 1958. 6. M. Berger and V. Mizel, Volterra equations with Itˆ o integrals, I, II, J. Int. Eqs. 2, 187–245; and 319–337 (1980). 7. D. A. Carlson, Infinite-horizon optimal controls for problems governed by a Volterra integral equation with state-and-control-dependent discount factor, J. Optim. Theory Appl. 66, 311–336 (1990). 8. S. Chen, X. Li, and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim. 36, 1685–1702 (1998). 9. S. Chen and J. Yong, Stochastic linear quadratic optimal control problems with random coefficients, Chinese Ann. Math. Ser. B 21, 323–338 (2000).

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10. S. Chen and J. Yong, Stochastic linear quadratic optimal control problems, Appl. Math. Optim. 43, 21–45 (2001). 11. S. Chen and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, II, SIAM J. Control Optim. 39, 1065–1081 (2000). 12. N. Dunford and J. T. Schwartz, Linear Operators, Part II, Spectral Theory, Selfadjoint Operators in Hilbert Space, John Wiley & Sons, Inc., New York, 1963. 13. D. Duffie and C.-F. Huang, Stochastic production-exchange equilibria, Research paper No. 974, Graduate School of Business, Stanford Univ, 1986. 14. R. E. Kalman, Contributions to the theory of optimal control, Bol. SOc. Mat. Mexicana 5, 102–119 (1960). 15. M. T. Kamien, and E. Muller, Optimal control with integral state equations, Rev. Economic Studies 43, 469–473 (1976). 16. A. M. Letov, Analytic design of regulators, Avtomat. i Telemekh, (1960), 436–446, 561–571, 661–669 (in Russian); English transl. in Automat. Remote Control, 21 (1960). 17. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, I. Abstract parabolic systems; II. Abstract hyperbolic-like systems over a finite time horizon, Encyclopedia of Mathematics and its Applications, 74, 75, Cambridge University Press, Cambridge, 2000. 18. X. Li and J. Yong, Optimal Control Theory for Infinite-dimensional Systems, Birkh¨ auser, Boston, 1995. 19. J. Lin, Adapted solution of a backward stochastic nonlinear Volterra integral equation, Stochastic Anal. Appl. 20, 165–183 (2002). 20. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. 21. N. G. Medhin, Optimal processes governed by integral equations, J. Math. Anal. Appl. 120, 1–12 (1986). 22. L. W. Neustadt and J. Warga, Comments on the paper “Optimal control of processes described by integral equations. I” by V. R. Vinokurov, SIAM J. Control 8, 572 (1970). 23. E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab. 18, 1635–1655 (1990). 24. A. J. Pritchard and Y. You, Causal feedback optimal control for Volterra integral equations, SIAM J. Control Optim. 34, 1874–1890 (1996). 25. P. Protter, Volterra equations driven by semimartingales, Ann. Probab. 13, 519–530 (1985). 26. V. R. Vinokurov, Optimal control of processes described by integral equations, I, II, III, Izv. Vysˇs. Uˇcebn. Zaved. Matematika, 7 (62) (1967), 21–33; 8 (63) (1967), 16–23; 9 (64) (1967), 16–25; (in Russian) English transl. in SIAM J. Control 7 (1969), 324–336; 337–345; 346–355. 27. J. Willems, Least squares stational optimal control and the algebraic Riccati equation, IEEE Trans. Auto. Control 16, 621–634 (1971). 28. W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control 6, 681–697 (1968). 29. W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd Edition, Springer-Verlag, 1979. 30. H. Wu and X. Li, A linear quadratic problem with unbounded control in Hilbert spaces, Diff. Int. Eqs. 13, 529–566 (2000). 31. D. Yao, S. Zhang, and X. Y. Zhou, Stochastic linear-quadratic control via primal-dual semidefinite programming, SIAM Rev. 46, 87–111 (2004). 32. J. Yong, A leader-follower stochastic linear quadratic differential game, SIAM J. Control Optim. 41, 4, 1015–1041 (2002).

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33. J. Yong, Backward stochastic Volterra integral equations and some related problems, Stoch. Proc. Appl., to appear. 34. J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. 35. Y. You, Quadratic integral games and causal synthesis, Trans. Amer. Math. Soc. 352, 2737–2764 (2000).

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Chap16-ChenZengjing

AN ADDITIVITY OF MAXIMUM EXPECTATIONS AND ITS APPLICATIONS

ZENGJING CHEN∗ Department of Mathematics Shandong University, Jinan 250100, P.R. China E-mail: [email protected] MATT DAVISON, MARK REESOR Department of Applied Mathematics The University of Western Ontario London, Canada YING ZHANG Department of Mathematics and Statistics Acadia University, Canada

In this paper, we explore an additivity of a class of maximum (minimum) expectations which come from the pricing of contingent claim in incomplete markets. We give examples to show this property can be used to calculate maximum (minimum) expectations and Choquet integral. Furthermore, we also explore its applications in partial differential equation (shortly PDE) and the pricing of the contingent claims in incomplete markets. Keywords: Maximum expectation, partial differential equation, backward stochastic differential equation; contingent claim, Malliavin calculus.

1. Introduction Fixed time horizon T > 0, let {Wt }0≤t≤T be a d-dimensional standard Brownian motion defined on a completed probability space (Ω, F, P ) and {Ft }0≤t≤T be the natural filtration generated by Brownian motion {Wt }0≤t≤T , that is, Ft = σ(Ws ; s ≤ t), we assume F = FT . To ease of exposition, in this paper, we assume d = 1 and Ω is a Wiener space, that is Ω := C0 (0, T ), the set of all continuous functions {ωt } defined on [0, T ] with w0 = 0 and for any ω ∈ Ω, Brownian motion Wt (ω) = ωt . We present the following notations that will be used in the rest of this

∗ Financial

support partly from NSF of China (10325106) and (10131030), FANEDD (2001059). 67

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paper. Let p ∈ (1, ∞) and set n o RT Lp (0, T ) := V : all Ft −adapted processes {Vt } such that E 0 |Vs |p ds < ∞ ; B 2 :=nL2 (0, T ) × L2 (0, T ); P := Q : all probability measures Q s.t. {E[ dQ dP |Ft ]} is a square integrable o martingale .

Lp (Ω, F, P ) := {ξ : all F−measurable random variables such that E|ξ|p < ∞} ;

We define maximum (minimum) expectations by E[ξ] := sup EQ [ξ],

E[ξ] := inf EQ [ξ];

(1)

E[ξ|Ft ] := ess inf EQ [ξ|Ft ].

(2)

Q∈P

Q∈P

and conditional expectations by E[ξ|Ft ] := ess sup EQ [ξ|Ft ],

Q∈P

Q∈P

In order to make maximum expectation E[ξ] be well-defined for any ξ ∈ Lp (Ω, F, P ), we further need some assumptions on P. Indeed, for any Q ∈ P, let Mt := E[ dQ dP |Ft ], because {Mt }0≤t≤T is a square integrable martingale, by Martingale Representation Theorem (see for example, Theorem 4.3.3, p. 53, Ref. 13), there exists a unique predictable process h ∈ L2 (0, T ) such that Z t Mt = 1 + hs dWs . 0

Let as :=

hs Ms ,

if E

RT 0

2

|as | ds < ∞, then Z t Mt = 1 + as Ms dWs . 0

Solving the above linear stochastic differential equation (shortly SDE), we obtain Mt is of the following form: 1

Mt = e − 2

Rt 0

|as |2 ds+

Rt 0

as dWs

, 0 ≤ t ≤ T.

Which implies that for Q ∈ P, there exists a process {at } such that form: RT RT 2 1 dQ = e− 2 0 as ds+ 0 as dWs . dP

dQ dP

is of the

(3)

We rewrite Q as Qa and call Qa the probability measure generated by {at }. Hence, P actually is the set of probability measures generated by some processes {a t } via (3). In this paper, we further assume there exists a deterministic positive function RT {k(t)} with 0 k 2 (s)ds < ∞ such that a R R − 12 0T as 2 ds+ 0T as dWs a dQ =e , |at | ≤ k(t), t ∈ [0, T ] . (∗) P= Q : dP

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Immediately, the maximum (minimum) expectations defined in (1) and (2) are welldefined. Indeed, for any Qa ∈ P and ξ ∈ Lp (Ω, F, P ), by Schwartz inequality and the fact that 1

E(e− 2

RT 0

|qas |2 ds+

RT 0

qas dWs

) = 1, ∀q ∈ (1, ∞),

we get a

|EQa ξ| ≤ E|ξ|| dQ dP | a

1

1

q q ≤ (E|ξ|p ) p (E| dQ dP | ) 1

1

≤ (E|ξ|p ) p (Ee− 2 1

q−1 2

1

q

= (E|ξ|p ) p e

≤ (E|ξ|p ) p e p where

1 p

+

1 q

RT

RT 0

RT

0

0

|qas 2 |ds+

|k(s)|2 ds

|k(s)|2 ds

RT 0

qas dWs + q

2 −q 2

RT 0

|as |2 ds

1

)q

< ∞,

= 1. Thus p

1

q

|E[ξ]| = | sup EQ ξ| ≤ (E|ξ| ) p e p Q∈P

RT 0

|k2 (s)|ds

< ∞,

which implies E[ξ] and E[ξ|Ft ] are well-defined for any ξ ∈ Lp (Ω, F, P ), so are E[ξ] and E[ξ|Ft ]. In this paper, we only consider maximum expectation E[·] as minimum expectation E[·] can be treated by the fact that E[ξ] = −E[−ξ],

E[ξ|Ft ] = −E[−ξ|Ft ], ∀ξ ∈ Lp (Ω, F, P ).

(∗∗)

To simplify notations, let us write E[·] and E[·|Ft ] simply as E[·] and E[·|Ft ] respectively. Obviously, by the definition of E[·], E[·] is a nonlinear operator on Lp (Ω, F, P ) and E[ξ + η] ≤ E[ξ] + E[η], ∀ξ, η ∈ Lp (Ω, F, P ), moreover, the equality holds if and only if P = {P }. We are interesting in finding some conditions on ξ and η under which the following relations hold. E[ξ + η] = E[ξ] + E[η], or E[ξ + η] < E[ξ] + E[η]. We will indicate the applications of the above results in Section 3. The maximum (minimum) expectations defined in (1) and (2) actually come from the pricing of contingent claim in incomplete markets. As we know, for given contingent claim ξ, which usually depends on the price of stocks, if the market is a complete market, then there exists a unique equivalent martingale measure Q such that the pricing of this contingent claim at time t is given by conditional expectation EQ [ξ|Ft ]. However, if the market is an incomplete market, such probability measure

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Q is not unique. Thus it is impossible to price and hedge this claim. In that case, an investor hopes to know the superpricing E[ξ] and subpricing E[ξ] (cf. El Karoui and Quenez10 ; Karatzas8; Cvitanic6 ; F¨ ollmer and Kramkov7). Hence, to explore the pricing of contingent claim in incomplete markets, it is important to study the properties of maximum expectations E[ξ] and minimum expectations E[ξ]. 2. Main Result In this paper, we will use backward stochastic differential equations (BSDE) and Malliavin calculus to study the properties of maximum expectations. We refer readers to Refs. 14 and 9 for BSDEs and Refs. 11 and 12 for Malliavin calculus. Let us now give the following Lemmas: Lemma 2.1. If ξ ∈ L2 (Ω, F, P ) and (yt , zt ) is the unique solution of BSDE, yt = ξ +

Z

T

k(s)|zs |ds − t

Z

T

zs dWs ,

(4)

t

where k(t) is defined in (∗), then E[ξ|Ft ] = yt ;

E[ξ] = y0 .

Proof. For any n > 0, let k (n) (t) = k(t) ∧ n and v R R (n) − 12 0T |vs |2 ds+ 0T vs dWs (n) v dQ , |v(t)| ≤ k (t) . =e P := Q : dP Obviously, P (n) ⊂ P (n+1) ⊂ · · · ⊂ P and P (n) ↑ P. Since |k (n) (t)| ≤ n, thus BSDE(5) below satisfies uniform Lipschitz condition. By Theorem 2.1 of Ref. 9, for each n > 0 , there exists a unique solution (yt (n) , zt (n) ) ∈ B 2 such that yt

(n)

=ξ+

Z

T

k

(n)

(s)|zs

(n)

|ds −

0

Z

T

zs (n) dWs .

(5)

0

Moreover, by Proposition 35.2 of Ref. 15, there exists a constant c > 0 such that Z T 2 E |zs (n) | ds ≤ c E|ξ 2 |. 0

(n)

Let E [ξ|Ft ] := ess supQ∈P (n) EQ [ξ|Ft ], by Proposition 3.2.2 of Ref. 10, or Lemma 1 in Ref. 4, E (n) [ξ|Ft ] = yt (n) . Note that E (n) [ξ|Ft ] ↑ E[ξ|Ft ], as n → ∞, it remains to prove that the limit of {yt (n) } is the solution of BSDE (4).

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In fact, for any n, m ≥ 1, by Proposition 35.2 of Ref. 15, there exists a constant C > 0 such that RT 2 2 E[sup0≤t≤T |yt (n) − yt (m) | + 0 |zs (n) − zs (m) | ds] h R i2 T (n) (n) ≤ C E 0 (|k (n) (s)|zs | − k (m) (s)|zs |)ds RT R T (n) ≤ C 0 |k (n) (s) − k (m) (s)|2 dsE 0 |zs |2 ds R T ≤ CcE|ξ|2 0 |k (n) (s) − k (m) (s)|2 ds → 0, as n, m → ∞, which implies that (y (n) , z (n) ) is a Cauchy sequence of B 2 , thus there exists (y, z) ∈ B 2 such that (y (n) , z (n) ) → (y, z), as n → ∞. Since R 2 T E 0 |k(s)|zs | − k (n) (s)|zs (n) ||ds 2 R RT T ≤ E 0 k(s)|zs (n) − zs |ds + E 0 |k (n) (s) − k(s)||zs (n) |ds RT RT RT RT . 2 2 ≤ 0 k 2 (s)dsE 0 |zs (n) − zs | ds + 0 |k (n) (s) − k(s)| dsE 0 |zs (n) |2 ds R R T (n) RT 2 2 2 T ≤ 0 k (s)dsE 0 |zs − zs | ds + cEξ 2 0 |k (n) (s) − k(s)| ds → 0, as n → ∞. Thus, Z

T

k

(n)

(s)|zs

(n)

|ds →

0

Z

T

k(s)|zs |ds,

as n → ∞

0

in L2 (Ω, F, P ). Set n → ∞ on both sides of BSDE (5), then (y, z) is the solution of BSDE(4). Thus yt = E[ξ|Ft ]. In particular, let t = 0, we get y0 = E[ξ]. Using the same method and the fact (∗∗), we can get Corollary 2.1. Assume ξ ∈ L2 (Ω, F, P ), then E[ξ|Ft ] is the unique solution of BSDE: Z T Z T yt = ξ − k(s)|zs |ds − zs dWs . 0

t

RT Lemma 2.2. Let ξ ∈ L2 (Ω, F, P ) be of the form ξ = f ( 0 σs dWs ), where f (x) is a continuous differentiable function, {σt } is a deterministic function. Let (yt , zt ) be the solution of the following BSDE Z T Z T yt = ξ + k(s)|zs |ds − zs dWs . (6) t

t

(i) If f is increasing (resp. decreasing), then σt zt ≥ 0, (resp. σt zt ≤ 0). (ii) Assume f is strictly increasing (resp. decreasing), if σt 6= 0, then zt 6= 0, a.e. t ∈ [0, T ].

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Proof. The proof of (i): For any ε > 0, ( |z|, gε (z) := 1 z2 2 (ε + ε ),

if |z| ≥ ε; if |z| < ε.

Obviously, gε (z) → |z|, as ε → 0. Let (yt (ε) , zt (ε) ) be the solution of BSDE Z T Z T Z T yt (ε) = f ( σs dWs ) + k(s)gε (zs )ds − zs (ε) dWs . t

0

t

(ε)

(ε)

Thus, by Proposition 35.2 of Ref. 15, (yt , zt ) → (y, z) in B 2 as ε → 0, where (y, z) is the solution of BSDE (6). Thanks to Proposition 5.3 of Ref. 9, we have for ε > 0, zt (ε) = Dt yt (ε) . where Dt yt (ε) denotes the Malliavin derivative of yt (ε) . Moreover, Dt yt (ε) satisfies the following linear BSDE Z T Z T Z T Dt yt (ε) = Dt f ( σs dWs ) + k(s)gε 0 (zs (ε) )Dt zs (ε) ds − Dt zs (ε) dWs , t

0

t

where gε (z) denotes the derivative of gε (z) with respect to z. Solving the above linear BSDE, we have Z T (ε) Dt yt = EQ [Dt f ( σs dWs )|Ft ], 0

0

dQ dP 0

− 12

RT

2

0

ε

2

RT

0

(ε)

[k (s)gε (zs )] ds+ 0 k(s)gε (zs )dWs 0 . =e where Let f be the derivative of f with respect to x, by the definition of Malliavin derivative, we have Z T Z T Dt f ( σs dWs ) = f 0 ( σs dWs )σt , 0

thus zt (ε) = Dt yt (ε) = EQ [f 0 (

0

Z

T

σs dWs )σt |Ft ] = σt EQ [f 0 ( 0

Z

T

σs dWs )|Ft ],

(7)

0

which implies if f 0 (x) ≥ 0, ∀x ∈ R, then σt zt (ε) ≥ 0 and σt zt (ε) ≤ 0 for f 0 (x) ≤ 0, ∀x ∈ R. It then follows by the fact z (ε) → z in L2 (0, T ) that we complete the proof of (i). The proof of (ii):R Without loss of generalization, we assume f 0 > 0, from T (ε) (i), σt zt = σt2 EQ [f 0 ( 0 σs dWs )|Ft ], note that |gε0 | ≤ 1, thus Q ∈ P, by strict RT (ε) Comparison Theorem 35.5 of Ref. 15, σt zt ≥ σt2 E[f 0 ( 0 σs dWs )|Ft ] > 0, the proof of (ii) is complete. RT RT p Let ξ, η ∈ L (Ω, F, P ) be of the forms ξ := f ( 0 σs dWs ), η := h( 0 vs dWs ), where {σt } and {vt } are two deterministic functions, f and h are two functions

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with continuous differentiation, under the above assumptions, We have the following main result. Theorem 2.1. Suppose that ξ and η are the random variables defined above, f and h are both increasing (decreasing) functions, (i) If σt vt ≥ 0, t ∈ [0, T ], then E[ξ + η] = E[ξ] + E[η]. (ii) If f and h are strictly increasing (decreasing) functions and σt vt < 0, t ∈ [0, T ], then E[ξ + η] < E[ξ] + E[η]. Proof. We turn our proof into three steps. Step 1. We assume f and h are bounded by some constant N > 0, i.e. |f | ≤ N, |h| ≤ N. Let yt = E[ξ|Ft ], y t = E[η|Ft ]. By Lemma 2.1, {yt } and {yt } are the solutions of BSDEs respectively, Z T Z T yt = ξ + k(s)|zs |ds − zs dWs (8) t

t

and yt = η +

Z

T

k(s)|z s |ds − t

Z

T

z s dWs .

(9)

t

By Lemma 2.2(i), we have σt zt ≥ 0, and vt z t ≥ 0,

a.e. t ∈ [0, T ].

It then follows by σt vt ≥ 0 that we have z t zt ≥ 0, which implies |z t | + |zt | = |z t + zt |,

a.e. t ∈ [0, T ].

Adding BSDE (8) with BSDE (9), we have Z T Z yt + y t = ξ + η + k(s)|zs + z s |ds − t

(10)

T

(zs + z s )dWs , t

which means (yt + y t , zt + z t ) is the solution of BSDE Z T Z T yt = ξ + η + k(s)|zs |ds − zs dWs , t

t

by Existence and Uniqueness Theorem of BSDE (Theorem 2.1 of Ref. 9) and Lemma 1, yt + yt = E[ξ + η|Ft ].

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That is, E[ξ|Ft ] + E[η|Ft ] = E[ξ + η|Ft ].

(11)

Step 2. For sufficient large N > 0 and N = 1, 2, · · · , let us denote (n)

fN (x) := xe−nd

(N )

(x)

,

(n)

hN (x) := xe−nd

(N )

(x)

(12)

,

where d(N ) (x) is defined by d

(N )

(x) :=

0, (x − N )1+α ,

|x| ≤ N ; |x| > N, 0 < α < 1.

(n)

(n)

It is easy to check that fN (x) → xI(|x|≤N ) and hN (x) → xI(|x|≤N ) as N → ∞, (n)

(n)

here and below IA is an indicator function. Obviously fN and hN are bounded and continuous differentiable. By Step 1, Z T Z T Z T (n) (n) (n) E[fN ( σs dWs ) + hN ( vs dWs )|Ft ] = E[fN ( σs dWs )|Ft ] 0

0

0

(n) + E[hN (

Z

T

vs dWs )|Ft ]. 0

(n) R T (n) R T Note that fN ( 0 σs dWs ) → ξ and hN ( 0 vs dWs ) → η as n, N → ∞ in Lp (Ω, F, P ) . We have

E[ξ + η|Ft ] = E[ξ|Ft ] + E[η|Ft ]. In particular, let t = 0, E[ξ + η] = E[ξ] + E[η]. Step 3. If f and g are strictly increasing (decreasing), by Lemma 2.2(ii) and the assumptions, we have σt vt < 0. Thus, z t zt < 0 , ∀t ∈ [0, T ]. Moreover, Equality (10) becomes |z t | + |zt | > |z t + zt |,

t ∈ [0, T ],

(13)

this with strict Comparison Theorem 35.5 of Ref. 15, we have E[ξ|Ft ] + E[η|Ft ] > E[ξ + η|Ft ]; t ∈ [0, T ]. Let t = 0, the proof is complete. Using the same method as the proof of Theorem 2.1, we can get the following Corollary: RT Corollary 2.2. Suppose that ξ = f ( 0 σs dWs ) is defined in Theorem 2.1, let a1 ≤ a2 ≤ · · · < an be a sequence, then E[

n X i=1

I(ξ≥ai ) ] =

n X i=1

E[Iξ≥ai ) ].

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3. Applications To illustrate the applications of the above main results, we give the following examples: 3.1. Compute maximum expectation In general, it is not easy to calculate maximum expectations, but for special case, we can calculate maximum expectations by using main results in this paper, let us give the following examples. RT RT Example 3.1. Compute supQ∈P EQ e−2 0 k(s)dWs − e2 0 k(s)dWs . In fact, let f (x) := e−2x and h(x) := −e2x , then f and h are monotone, by Theorem 2.1, RT RT RT RT supQ∈P EQ e−2 0 k(s)dWs − e2 0 k(s)dWs = E[e−2 0 k(s)dWs ] + E[−e2 0 k(s)dWs ] RT

= E[e−2

By Lemma 2.1, E[e−2 BSDE at time t = 0,

RT 0

k(s)dWs

yt = e−2

RT 0

0

k(s)dWs

] − E[e2

RT 0

k(s)dWs

].

] is the value of (yt ), the solution of the following

k(s)dWs

+

Z

T

k(s)|zs |ds − t

Z

T

zs dWs . t

Solving the above BSDE, we get Rt Rt (yt , zt ) = e−2 0 k(s)dWs , 2k(t)e−2 0 k(s)dWs . RT

Thus E[e−2 0 k(s)dWs ] = y0 = 1. Similarly, by Corollary 2.1, E[e2 value of {yt }, the solution of the following BSDE, at time t = 0, Z T Z T R 2 0T k(s)dWs yt = e − k(s)|zs |ds − zs dWs . t

RT 0

k(s)dWs

] is the

t

Solving the above BSDE, we obtain Rt Rt (yt , zt ) = e2 0 k(s)dWs , −2k(t)e2 0 k(s)dWs , and E[e2

RT 0

k(s)dWs

] = 1. Thus E[e−2

RT

k(s)dWs

0

− e2

RT 0

k(s)dWs

] = 1 − 1 = 0.

Remark 3.1. From this example, we get the following interesting facts: (1) Since RT 2 RT Ee−2 0 k (s)ds−2 0 k(s)dWs = 1, thus Ee−2

RT 0

k(s)dWs

= e2

which depends on k, but by Example 3.1, E[e−2

RT 0

k2 (s)dWs

RT 0

k2 (s)ds

,

] = 1,

which does not depend on k. R RT −2 0T k(s)dWs (2) Since E[e ] = E[e2 0 k(s)dWs ] = 1, thus for any Q ∈ P, EQ [e−2

RT 0

k(s)dWs

] ≤ EQ [e2

RT 0

k(s)dWs

].

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3.2. Compute Choquet integral For any A ∈ F, let V (A) := supQ∈P Q(A). Obviously V (·) is a capacity and V (A) = E[IA ]. Choquet5 introduced the following integral (shortly Choquet expectation) which generalizes the usual mathematical expectations, Z 0 Z ∞ C[ξ] := (V (ξ ≥ t) − 1)dt + V (ξ ≥ t)dt. (14) 0

−∞

Example 3.2. Compute C[e−2 Let ξ := e−2

RT 0

k(s)dWs

RT 0

k(s)dWs

].

, for sufficient large N and n = 1, 2, · · · , and n

(n) ξN

:=

2 X iN i=0

2n

, I( iNn ≤ξ∧N < (i+1)N ) n 2

2

(n)

then ξN → ξ as N, n → ∞ in L2 (Ω, F, P ). Note that ξ ∧ N ≤ N and P2n iN P2n iN iN I = I − I (i+1)N (i+1)N iN n n (ξ∧N ≥ 2n ) i=0 2 i=0 2 ( 2n ≤ξ∧N < 2n ) (ξ∧N ≥ 2n ) P2n iN P2n +1 (i−1)N = i=1 2n I(ξ∧N ≥ iNn ) − i=2 2n I(ξ∧N ≥ iN 2 2n ) P2n iN P2n (i−1)N = i=1 2n I(ξ∧N ≥ iNn ) − i=2 2n I(ξ∧N ≥ iNn ) 2 2 n N P2 = 2n i=1 I(ξ∧N ≥ iNn ) . 2

Applying Corollary 2.2, n

E[

n

2 X iN i=0

2 N X I E[I(ξ∧N ≥ iNn ) ] (i+1)N ] = iN 2 2n ( 2n ≤ξ∧N < 2n ) 2n i=1

We have, from the definition (14) and applying the fact that V (ξ ∧ N ≥ t) = E[I(ξ∧N ≥t) ], C[e−2

RT 0

k(s)dWs

]= = = = =

RT V e−2 0 k(s)dWs > t dt RN limN →∞ 0 V (ξ ∧ N ≥ t)dt P 2n limN →∞ limn→∞ i=0 2Nn V (ξ ∧ N ≥ iN 2n ) P 2n N limN →∞ limn→∞ i=0 2n E[I(ξ∧N ≥ iNn ) ] 2 P2n iN limN →∞ limn→∞ E[ i=0 2n I( iNn ≤ξ∧N ≤ (i+1)N ] ) n R∞ 0

2

= limN →∞ E[ξ ∧ N ] = E[e−2 According to Example 3.1, E[e−2

RT 0

k(s)dWs

RT 0

2

k(s)dWs

] = 1, thus C[e−2

RT 0

].

k(s)dWs

] = 1.

Remark 3.2. This example also shows that under some assumptions on ξ, the maximum expectation of ξ is equal to Choquet expectation.

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3.3. Application to PDE Suppose that σt , bt , K(t), Φ are continuous and bounded functions. Let UΦ be the solution of the following PDE with respect to terminal value Φ, ( ∂U 1 2 ∂2U ∂U ∂U ∂t + 2 σt ∂x2 + bt ∂x = k(t)| ∂x σt x|; (15) U (T, x) = Φ(x), x ∈ R+ , 0 ≤ t ≤ T. Obviously, In general, UΦ1 +Φ 6= UΦ1 + UΦ2 . However, we have Example 3.3. Assume Φ1 and Φ2 are two continuous bounded functions, UΦ1 and UΦ2 are the solution of PDE (15) corresponding to Φ = Φ1 and Φ = Φ2 . If Φ is an increasing or decreasing function, then UΦ1 +Φ2 = UΦ1 + UΦ2 , that is, the solution of PDE (15) is linear with respect to Φ. In fact, by Theorem 12.3 of Ref. 1, UΦ (t, x) = yt t,x , where {ys t,x }0≤s≤T is the solution of BSDE yt t,x = Φ(XT t,x ) +

Z

T

k(s)|zs t,x |ds −

t

Z

T

zs t,x dWs .

t

And {Xs t,x } is the solution of SDE

dXs t,x = bt Xs ds + σs Xs dWs Xt = x, t ≤ s ≤ T.

By Lemma 1 and Proposition 4.2 of Ref. 9, yt t,x = E[Φ(XT t,x )|Ft ] = E[Φ(XT t,x )]. That is, UΦ (x, t) = E[Φ(XTt,x )]. Thus if Φ1 and Φ2 are increasing (decreasing) functions with continuous differentiation, applying Theorem 2.1, we have E[Φ1 (XTt,x ) + Φ2 (XTt,x )] = E[Φ1 (XTt,x )] + E[Φ2 (XTt,x )]. That is, UΦ1 +Φ2 = UΦ1 + UΦ2 .

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3.4. Application to European Option As explained in introduction, maximum expectations defined in this paper come from the pricing of contingent claim in incomplete market, for given contingent claims ξ and η, the superpricing of contingent claims given the maximum expectation E[ξ] and E[η], since E[ξ + η] ≤ E[ξ] + E[η], thus, in an incomplete market, it is better for an investor to buy ξ and η together than to buy them separately. Let us now further consider the application of our result in incomplete markets. If given two contingent claims XT and YT which satisfy the following SDEs dXt = at Xt dt + σt Xt dWt ; X0 = x > 0, 0 ≤ t ≤ T. and

dYt = bt Yt + vt Yt dWt ; Y0 = y > 0, 0 ≤ t ≤ T.

Where at , bt , σt , vt are continuous bounded functions. the above linear YTR are of Rthe forms XT = R T SDEs, we get XT and − R Solving T T 2 1 T f ( t σs dWs ) and YT = h( 0 vs dWs ). where f (x) = e 0 at dt− 2 0 σt dt+x , h(x) = RT R T 2 1 e− 0 bt dt− 2 0 vt dt+x . The maximal pricing for investors to buy XT and YT together is E[YT + XT ]. However, by Theorem 2.1, if σt vt ≥ 0, then E[YT + XT ] = E[YT ] + E[XT ], which implies that the maximal price that the investors buy XT and YT together is equal to buy them separately. However, if σt vt < 0, then E[YT + XT ] < E[YT ] + E[XT ]. In this case, the investors should buy the two contingent claims together to hedge risk. References 1. E. Barles and E. Lesigne, SDE, BSDE and PDE, Pitman Research Notes in Mathematics Series No. 364, 47–80 (1997). 2. Z. Chen, A property of backward stochastic differential equations, C.R. Acad. Sci. Paris No. 1, 483–488 (1998). 3. Z. Chen and L. Epstein, Ambiguity, risk and asset return in continuous time, Econometrica 70, 1403–1443 (2002). 4. Z. Chen and S. Peng, A general downcrossing inequality for g-martingales, Statist. Probab. Lett. 46, no. 2, 169–175 (2000). 5. G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5, 131–295 (1955).

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6. J. Cvitanic, Minimizing expected loss of hedging in incomplete and constrained markets, Preprint, Columbia Univ. New York (1998). 7. H. F¨ ollmer and D. Kramkov, Optimal decomposition under constraints, Probab. Theory & Rel. Fields 109, 1–25 (1997). 8. I. Karatzas, Lectures in mathematical finance, Providence, American Mathematical Society (1997). 9. N. El Karoui, S. Peng, and M. Quenez, Backward stochastic differential equations in finance, Math. Finance No. 1, 1–71 (1997). 10. N. El Karoui, M. Quenez, Dynamic programming and pricing contingent claims incomplete markets, SIAM J. Control Optim. 33, 29–66 (1995). 11. D. Nualart, The Malliavin Calculus and Related Topics, New York and Berlin, Springer-Verlag (1995). 12. B. ∅ksendal, An introduction to Malliavin Calculus with application to economics, Working paper No. 3/96, Norwegian School of Economics (1997). 13. B. ∅ksendal, Stochastic differential equations, 5th Edition, Springer (1991). 14. E. Pardoux and S. Peng, Adapted solution of a Backward stochastic differential equation, Systems and Control Letters 14, 55–61 (1990). 15. S. Peng, BSDE and related g–expectation, Pitman Research Notes in Mathematics Series No. 364, 141–159 (1997). 16. S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type, Probab. Theory & Rel. Fields 113, 473–499 (1999).

July 17, 2007

10:55


Chap17-HuYing

STOCHASTIC CONTROL AND BSDES WITH QUADRATIC GROWTH

MARCO FUHRMAN Dipartimento di Matematica, Politecnico di Milano piazza Leonardo da Vinci 32, 20133 MILANO, Italy E-mail: [email protected] YING HU IRMAR, Université Rennes 1 Campus de Beaulieu, 35042 RENNES Cedex, France E-mail: [email protected] GIANMARIO TESSITORE Dipartimento di Matematica, Universit` a di Parma Parco Area delle Scienze, 53/A - 43100 Parma, Italy E-mail: [email protected]

In this talk, we study a stochastic optimal control problem where the drift term of the equation has a linear growth on the control variable, the cost functional has a quadratic growth and the control process takes values in a closed set. This problem is related to some BSDE with quadratic growth. We prove that the optimal feedback control exists and the optimal cost is given by the initial value of the solution of the related backward stochastic differential equation.

1. Introduction This talk is based upon the paper 6 . The interested readers can see the full paper for all the proofs of the results presented here. In this talk, we consider a controlled equation of the form:

dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , ut ) dt], X0 = x.

t ∈ [0, T ],

6

(1)

In the equation, W is a Rd -valued Wiener process, defined on a complete probability space (Ω, F, P) with respect to a filtration F := (Ft )t≥0 satisfying the usual conditions; the unknown process X takes values in Rn ; x is a given element of Rn ; u is the control process, which is assumed to be an Ft -adapted process taking values in a given nonempty closed set K ⊂ Rm . The control problem consists in minimizing 80

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a cost functional of the form J =E

Z

T

g(t, Xt , ut ) dt + E φ(XT ).

(2)

0

We suppose that r has a linear growth in u, g has quadratic growth in x and u, and φ has quadratic growth in x. The main novelty of the present work, in comparison with the existing literature is that, on the one hand we neither assume K nor r to be bounded, on the other hand we consider a degenerate control problem (since nothing is assumed on the image of σ). Moreover we also allow φ to have linear growth. Nonlinear Backward Stochastic Differential Equations (BSDEs) were first introduced by Pardoux and Peng 9 . They have found applications in stochastic control, see, e.g., Refs. 3 and 10. To have an overview of recent applications of backward stochastic differential equations techniques to control problems being by several aspects more general than the one considered here but involving “bounded control image” assumptions see for instance Ref. 4 or Ref. 7 and references therein. The special “unbounded” case corresponding to the assumptions K = Rm and g = 21 |u|2 + q(t, x) is treated in Ref. 5 by an ad-hoc exponential transform. We notice that in Ref. 5 φ is allowed to take value +∞. Finally this same special case (in which the Hamiltonian is exactly the square of the norm of the gradient) was treated in Ref. 8 by analytic techniques, under non-degeneracy assumptions and in an infinite dimensional framework. The difficulty here is that the Hamiltonian corresponding to the control problem has quadratic growth in the gradient and the terminal cost is not bounded. Thus the backward stochastic differential equation corresponding to the problem has quadratic growth in the Z variable and unbounded terminal value. To treat such equation we have to apply the localization procedure recently introduced by Briand and Hu in Ref. 2. We notice that for such BSDEs no general uniqueness results are known: we replace uniqueness by the selection of a maximal solution. Moreover the usual application of Girsanov technique is not allowed (since the Novikov condition is not guaranteed) and we have to develop specific arguments both to prove the fundamental relation, see §4, and to obtain the existence of a (weak) solution to the closed loop equation, see §5. Our main result is to prove that the optimal feedback control exists and the optimal cost is given by the value Y0 of the maximal solution (Y, Z) of the backward stochastic differential equation (BSDE) with quadratic growth and unbounded terminal value mentioned above. Moreover we show that we can construct an optimal feedback in terms of the process Z. Finally we prove that if we fix a particular optimal feedback law then the solution of the corresponding closed loop equation is unique, see Proposition 5.2. The talk is organized as follows: in the next section, we describe the control problem; in section 3, we study the related BSDE; in section 4, we establish the fundamental relation between the optimal control problem and BSDE; and the last section is devoted to the proof of the existence of optimal feedback control.

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2. The Controlled Problem We consider here the optimal control problem given by a state equation of the form: dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , ut ) dt], t ∈ [0, T ], (3) X0 = x, and by a cost functional of the form Z T J =E g(t, Xt , ut ) dt + E φ(XT ).

(4)

0

We work under the following assumptions. Hypothesis A. (1) The process W is a Wiener process in Rd , defined on a complete probability space (Ω, F, P) with respect to a filtration (Ft ) satisfying the usual conditions. (2) The set K is a nonempty closed subset of Rm . (3) The functions b : [0, T ] × Rn → Rn , σ : [0, T ] × Rn → Rn×d , r : [0, T ] × Rn × K → Rd , g : [0, T ] × Rn × K → R, φ : Rn → R, are Borel measurable. (4) For all t ∈ [0, T ], x ∈ Rn , r(t, x, ·) and g(t, x, ·) are continuous functions from K to Rd and from K to R respectively. (5) There exists a constant C > 0 such that for every t ∈ [0, T ], x, x0 ∈ Rn , u ∈ K it holds: |b(t, x) − b(t, x0 )| ≤ C|x − x0 |,

|b(t, x)| ≤ C(1 + |x|),

(5)

|σ(t, x) − σ(t, x0 )| ≤ C|x − x0 |,

(6)

|σ(t, x)| ≤ C,

(7)

|r(t, x, u) − r(t, x0 , u)| ≤ C(1 + |u|)|x − x0 |,

(8)

|r(t, x, u)| ≤ C(1 + |u|),

(9)

0 ≤ g(t, x, u) ≤ C(1 + |x|2 + |u|2 ),

(10)

0 ≤ φ(x) ≤ C(1 + |x|2 ).

(11)

(6) There exist R > 0 and c > 0 such that for every t ∈ [0, T ], x ∈ Rn , and every u ∈ K satisfying |u| ≥ R g(t, x, u) ≥ c|u|2 .

(12)

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We will say that an (Ft )-adapted stochastic process {ut , t ∈ [0, T ]} with values in K is an admissible control if it satisfies: Z T E |ut |2 dt < ∞. (13) 0

This square summability requirement is justified by (12): a control process which is not square summable would have infinite cost. Next we show that for every admissible control the solution to (3) exists. Proposition 2.1. Let u be an admissible control. Then there exists a unique continuous, (Ft )-adapted process X satisfying E supt∈[0,T ] |Xt |2 < ∞ and, P-a.s, Z t Z t Z t Xt = x+ b(s, Xs ) ds+ σ(s, Xs ) dWs + σ(s, Xs ) r(s, Xs , us ) ds, t ∈ [0, T ]. 0

0

0

Proof. The proof of Proposition 2.1 relies on an approximation procedure. The stochastic control problem associated with (3)-(4) consists in minimizing the cost functional J(x, u) among all the admissible controls. 3. The Forward-Backward System Let us consider again the functions b, σ, g, φ satisfying the assumptions in Hypothesis A. We define the Hamiltonian function ψ(t, x, z) = inf [g(t, x, u) + z · r(t, x, u)], u∈K

t ∈ [0, T ], x ∈ Rn , z ∈ Rd ,

(14)

where · denotes the usual scalar product in Rd . We collect some immediate properties of the function ψ. Lemma 3.1. The map ψ is a Borel measurable function from [0, T ] × Rn × Rd to R. There exists a constant C > 0 such that −C(1 + |z|2 ) ≤ ψ(t, x, z) ≤ g(t, x, u) + C|z|(1 + |u|),

∀u ∈ K.

(15)

Moreover the infimum in (14) is attained in ball of radius C(1 + |x| + |z|) that is ψ(t, x, z) =

min

[g(t, x, u)+z·r(t, x, u)],

u∈K,|u|≤C(1+|x|+|z|)

t ∈ [0, T ], x ∈ Rn , z ∈ Rd , (16)

and ψ(t, x, z) < g(t, x, u) + z · r(t, x, u) if |u| > C(1 + |x| + |z|).

(17)

Finally, for every t ∈ [0, T ] and x ∈ Rn , z → ψ(t, x, z) is continuous on Rd . Next we take an arbitrary complete probability space (Ω, F, P◦ ) and a Wiener process W ◦ in Rd with respect to P◦ . We denote by (Ft◦ ) the associated Brownian filtration, i.e. the filtration generated by W ◦ and augmented by the P◦ -null sets of F; (Ft◦ ) satisfies the usual conditions.

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We introduce the forward equation: dXt = b(t, Xt ) dt + σ(t, Xt ) dWt◦ , X0 = x,

t ∈ [0, T ],

(18)

whose solution is a continuous (Ft◦ ) adapted process, which exists and is unique by classical results. Next we consider the associated backward equation dYt = −ψ(t, Xt , Zt ) dt + Zt dWt◦ , t ∈ [0, T ], (19) YT = φ(XT ). The solution of (19) exists in the sense specified by the following proposition. Proposition 3.1. Assume that b, σ, g, φ satisfy Hypothesis A. Then there exist Borel measurable functions v : [0, T ] × Rn → R,

ζ : [0, T ] × Rn → Rd

with the following property: for arbitrary complete probability space (Ω, F, P ◦ ) and Wiener process W ◦ in Rd , denoting by X the solution of (18), the processes Y, Z defined by Yt = v(t, Xt ),

Zt = ζ(t, Xt )

satisfy E

◦

2

sup |Yt | < ∞,

E

◦

t∈[0,T ]

Z

T

|Zt |2 dt < ∞, 0

Y is continuous, nonnegative, and finally, P -a.s, Z T Z T ◦ Zs dWs = φ(XT ) + ψ(s, Xs , Zs ) ds, Yt + ◦

t

t ∈ [0, T ].

t

Proof. We apply the localization procedure together with a priori bounds, which is introduced by Briand and Hu 2 .

4. The Fundamental Relation In this section we revert to the notation introduced in the first section. Proposition 4.1. For every admissible control u and for the corresponding trajectory X starting at x we have Z T J(u) = v(0, x) + E [−ψ(t, Xt , ζ(t, Xt )) + ζ(t, Xt ) · r(t, Xt , ut ) + g(t, Xt , ut )] dt. 0

Proof. We apply an approximation procedure together with the Girsanov theorem.

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Corollary 4.1. For every admissible control u and any initial datum x we have J(u) ≥ v(0, x) and the equality holds if and only if the following feedback law holds P-a.s. for a.e. t ∈ [0, T ] ψ(t, Xt , ζ(t, Xt )) = ζ(t, Xt ) · r(t, Xt , ut ) + g(t, Xt , ut ), where X is the trajectory starting at x and corresponding to control u. 5. Existence of Optimal Controls: The Closed Loop Equation Let us consider again the functions b, σ, g, φ satisfying the assumptions in Hypothesis A. We recall the definition of the Hamiltonian function t ∈ [0, T ], x ∈ Rn , z ∈ Rd .

ψ(t, x, z) = inf [g(t, x, u) + z · r(t, x, u)], u∈K

(20)

Lemma 5.1. There exists a Borel measurable function γ : [0, T ] × Rn × Rd → K such that ψ(t, x, z) = g(t, x, γ(t, x, z)) + z · r(t, x, γ(t, x, z)),

t ∈ [0, T ], x ∈ Rn , z ∈ Rd . (21)

Moreover there exists a constant C > 0 such that |γ(t, x, z)| ≤ C(1 + |x| + |z|).

(22)

Proof. It suffices to apply the Filippov theorem 1 . Next we address the problem to find a weak solution to the so-called closed loop equation. We define u(t, x) = γ(t, x, ζ(t, x)),

t ∈ [0, T ], x ∈ Rn ,

where ζ has been introduced in Proposition 3.1. The closed loop equation is dXt = b(t, Xt ) dt + σ(t, Xt ) [dWt + r(t, Xt , u(t, Xt )) dt], t ∈ [0, T ], (23) X0 = x. By a weak solution we mean a complete probability space (Ω, F, P) with a filtration (Ft ) satisfying the usual conditions, a Wiener process W in Rd with respect to P and (Ft ), and a continuous (Ft )-adapted process X with values in Rn satisfying, P-a.s., Z T |u(t, Xt )|2 dt < ∞, (24) 0

and We note that by (9) it also follows that R T such that (23) holds. |r(t, X , u(t, X ))| dt < ∞, P-a.s., so that (23) makes sense. t t 0

Proposition 5.1. Assume that b, σ, g, φ satisfy Hypothesis A. Then there exists a weak solution of the closed loop equation, satisfying in addition Z T E |u(t, Xt )|2 dt < ∞. (25) 0

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Corollary 5.1. By Corollary 4.1 it immediately follows that if X is the solution to (23) and we set u]s = u(s, Xs ) then J(x, u] ) = v(0, x) and consequently X is an optimal state, u]s is an optimal control and u is an optimal feedback. Next we prove uniqueness in law for the closed loop equation. We remark that the condition (24) is part of our definition of a weak solution. Proposition 5.2. Assume that b, σ, g, φ satisfy Hypothesis A. Fix γ : [0, T ] × Rn × Rd → K satisfying (21) (and consequently (22)) and let u(t, x) = γ(t, x, ζ(t, x)). Then the weak solution of the closed loop equation (23) is unique in law. References 1. J.-P. Aubin, H. Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2. Birkh¨ auser Boston Inc., Boston, MA, 1990. 2. P. Briand, Y. Hu, BSDE with quadratic growth and unbounded terminal value. Preprint no. 05-07, IRMAR, Université Rennes 1, 2005. 3. N. El Karoui, S. Peng, M. C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997). 4. N. El Karoui, S. Hamadène, BSDEs and risk-sensitive control, zero-sum and nonzerosum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107, 145–169 (2003). 5. M. Fuhrman, A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes. Stochastic Process. Appl. 108, 263-298 (2003). 6. M. Fuhrman, Y. Hu, G. Tessitore, On a class of stochastic optimal control problems related to BSDEs with quadratic growth. Preprint. 7. M. Fuhrman, G. Tessitore, Existence of optimal stochastic controls and global solutions of forward-backward stochastic differential equations. SIAM J. Control Optim. 43, 813–830 (2004). 8. F. Gozzi, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl. 198, 399-443 (1996). 9. E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Systems and Control Lett. 14, 55-61 (1990). 10. J. Yong, X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York, 1999.

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A FUNDAMENTAL THEOREM OF ASSET PRICING IN CONTINUOUS TIME WITH SQUARE INTEGRABLE PORTFOLIOS∗

HANQING JIN and XUN YU ZHOU Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong {hqjin, xyzhou}@se.cuhk.edu.hk

This paper studies a continuous-time market with multiple stocks whose prices are governed by geometric Brownian motions, and admissible investment portfolios are defined via certain square integrability condition. It is proved that, when the investment opportunity set is deterministic (albeit possibly time varying), such a market being arbitrage free is equivalent to the existence of a square integrable (in time) market price of risk, and as a result equivalent to the existence of an equivalent martingale measure. Counterexamples are given to show that these equivalent results are no longer true in a market with a stochastic investment opportunity set.

Keywords: Continuous-time financial market, arbitrage opportunity, FTAP, martingale, square integrability, equivalent martingale measure.

1. Introduction Arbitrage-free is a basic assumption and a starting point for various problems (e.g., portfolio selection, option pricing, etc.) associated with a financial market. The so-called fundamental theorem of asset pricing (FTAP) is about an equivalent condition of the absence of arbitrage, which, in addition to economically characterizing this fundamental property of a market, should also provide an easily verifiable mathematical criterion. This, in turn, would make some powerful mathematical tools (such as martingale theory and functional analysis) available to analyze the market. In a dynamic multi-period market, it is well known that arbitrage-free is equivalent to the existence of an equivalent martingale measure [see, e.g., Harrison and Kreps,10 Dalang, Morton and Willinger,4 and F¨ ollmer and Schied9 (p. 217, Theorem 5.17)]. This is a very elegant and neat result. When it comes to the continuous-time setting, however, the validity of FTAP becomes a subtle issue. As elaborated in the very informative introduction section of Ref. 5, in a continuous∗ Supported

by the RGC Earmarked Grants CUHK 4175/03E and CUHK418605. The authors are indebted to Jia-an Yan, Chinese Academy of Sciences, for his helpful comments on an earlier version of the paper. 87

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time market arbitrage-free is only “essentially” equivalent to the existence of an equivalent martingale measure. The precise meaning of the word “essential” depends on the specific mathematical model under consideration and, in particular, on how the class of admissible portfolios is defined (more on this later), which has led to a few variants of the continuous-time FTAP in the literature. In Ref. 5, a very general bounded or locally bounded semimartingale model for (discounted) securities prices is investigated, and a condition stronger than no-arbitrage, called no free lunch with vanishing risk (NFLVR), is introduced. It is proved that NFLVR is equivalent to the existence of an equivalent martingale or local martingale (depending on whether the underlying securities prices are bounded or locally bounded) measure. In Ref. 6, a sequel to Ref. 5, unbounded securities price processes are considered, and similar equivalent condition is given with the local martingale replaced by the so-called sigma-martingale. While the results in these two papers5, 6 are very deep, the technical analyses therein are extremely involved, largely due to the generality of the model. Moreover, the equivalent conditions derived are not for no-arbitrage itself; and the existence of the equivalent local martingale/sigmamartingale is a rather stiff mathematical condition, which seems to be very hard to verify and use. A more specific and widely used continuous-time market model is the so-called geometric Brownian motion (GBM), up to the presence of some additional features such as a dividend and/or consumption; see, e.g., Refs. 17 and 13. In this model, the financial market consists of a risk-free bond and multiple risky stocks whose price processes are driven by a (multi-dimensional) Brownian motion. In Ref. 13 (p. 12, Theorem 4.2), it is stated that a necessary condition for the absence of arbitrage is the existence of a market price of risk; yet no further properties of this market price of risk are specified. Consequently this necessary condition is far too weak to be sufficient. Sufficient conditions are indeed presented in Ref. 13, namely the market price of risk is square integrable in time almost surely and the pricing kernel is a martingale. These conditions are relatively easier to check; unfortunately they are not if-and-only-if conditions for the absence of arbitrage. This drawback is nicely overcome in Ref. 15, where a necessary and sufficient condition for no-arbitrage is derived for a Brownian motion driven market with an invertible volatility matrix, via a pure probabilistic approach. [The results were later extended to the semimartingale setting in Ref. 19.] There is a very important point we need to note when studying FTAP. By its very definition whether a market is arbitrage free or not depends critically on the set of admissible investment strategies (portfolios) adopted. The admissible portfolios under consideration in Refs. 5, 6, 13 and 15 are the so-called tame strategies, namely, the ones whose corresponding wealth processes are almost surely bounded from below (the lower bounds may be different with different portfolios). As painstakingly argued in Refs. 5 and 13, imposing a lower bound on the wealth is to rule out the notorious doubling strategy that will lead to a sure win even on a finite time hori-

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zon. However, if one must rule out the doubling strategy a priori, there are many ways to do so. Being unbounded from below is just one property of the doubling strategy; one could certainly impose different restrictions to prevent the doubling strategy from being a legitimate strategy. For instance, square integrable portfolios will also exclude the doubling strategy (see Remark 2.4). It is important to note that tame portfolios eliminate many meaningful investment strategies from consideration. As an easy example, in a Black–Scholes market with one bond and one stock, shorting one share of the stock is a perfectly reasonable portfolio, which is however not tame. In this paper, we consider a GBM market where admissible portfolios are those that are square integrable — without any uniform boundedness restriction. The square integrability of portfolios is widely adopted in both multi-period and continuous-time setups; see, e.g., Refs. 14 and 7. Yet to our best knowledge an equivalent no-arbitrage condition is still absent for this class of admissible portfolios associated with a GBM model. Our direct motivation to solve this open problem, though, arises from the recent extensive study on the continuous-time version of Markowitz’s Nobel-prize-winning mean–variance portfolio selection problem [see, e.g., Refs. 20, 16, 12 and 1]. In these works, the no-arbitrage property of the underlying market is a prerequisite. One should note that the very nature of the mean–variance model requires an admissible portfolio to be square integrable, for otherwise the variance of the corresponding terminal wealth would not even be defined. Therefore, a characterization of the no-arbitrage condition with square integrable portfolios will lay a foundation to the continuous-time mean–variance portfolio selection model. The main result of this paper is that, when the set of the market coefficients [aka investment opportunity set in Merton’s terminology 17 ] is deterministic (yet possibly time varying), arbitrage-free is equivalent to the existence of a square integrable (in time) market price of risk. As an immediate consequence, arbitrage-free is equivalent to the existence of an equivalent martingale measure. Therefore, we have a version of continuous-time FTAP that is as clean and neat as that in the discrete-time case. The result is derived via exploring some property of the space L2 (0, T ; IRm ) (Lemma 4.1), together with a series of delicate analyses on functions that are out of this space. Furthermore, we show through two counterexamples that these equivalent results cannot be extended to the case with a stochastic investment opportunity set in either directions. Clearly our results are not covered by those in Refs. 5 and 6 or Ref. 15 in that different sets of investment strategies are being considered, and we do have an equivalent condition for arbitrage-free, rather than for NFLVR. Meanwhile we believe that our approach is new and different from those in existing literature. In addition our proofs are constructive in that we show explicitly arbitrage policies when the (mathematical) equivalent condition fails.

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The remainder of this paper is organized as follows. The market model is set up in section 2, and some preliminary results are given in section 3. Section 4 presents the main result (FTAP) of the paper, for the case when the investment opportunity set is deterministic. Section 5 contains two counterexamples demonstrating that the main results are no longer true with a stochastic investment opportunity set. Section 6 is devoted to the issue of market completeness. Finally, section 7 concludes the paper.

2. The Financial Market Throughout this paper T > 0 is a fixed, finite terminal time, (Ω, F, P, {Ft }0≤t≤T ) is a filtered complete probability space on which a standard m-dimensional Brownian motion W (t) ≡ (W 1 (t), · · · , W m (t))T with W (0) = 0 is defined. It is assumed that the filtration {Ft : 0 ≤ t ≤ T } is generated by the Brownian motion and augmented by all the P -null sets. For −∞ ≤ a < b ≤ +∞ and 1 ≤ p < +∞ we denote by Lp (a, b; IRd ) the set of all IRd -valued, measurable functions f (·) such Rb that a |f (t)|p dt < +∞, by LpF (a, b; IRd ) the set of all IRd -valued, Ft -progressively Rb measurable stochastic processes f (·) = {f (t) : 0 ≤ t ≤ T } such that E a |f (t)|p dt < +∞, and by LpFT (IRd ) the set of all IRd -valued, FT -measurable random variables d d ∞ η such that E|η|p < +∞. Moreover, L∞ (a, b; IRd ), L∞ F (a, b; IR ), and LFT (IR ) respectively denote the spaces of essentially bounded functions, Ft -progressively measurable processes, and random variables. Furthermore, we use AT to denote the transpose of a matrix A. Finally, a.s. (the abbreviation of “almost surely”) signifies that the corresponding statement holds true with probability 1 (with respect to P ). Consider a market where n+1 assets are traded continuously. One of the assets is the bond whose price process S0 (t) is subject to the following (stochastic) ordinary differential equation: dS0 (t) = r(t)S0 (t)dt, S0 (0) = s0 > 0,

(1)

where the interest rate process r(·) is a scalar-valued, Ft -progressively measurable stochastic process. The other n assets are stocks whose price processes S i (t), i = 1, · · · , n, satisfy the following stochastic differential equations:

dSi (t) = Si (t) bi (t)dt +

m X j=1

σij (t)dW j (t) , Si (0) = si > 0,

(2)

where bi (·) and σij (·), the appreciation and volatility rate processes, respectively, are scalar-valued, Ft -progressively measurable stochastic processes. The set {r(·), bi (·), σij (·), i = 1, · · · , n, j = 1, · · · , m} is called the investment opportunity set.

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Consider an agent, with an initial capital x0 at time 0, has a total wealth x(t) at time t > 0, while the dollar amount invested in stock i, i = 1, · · · , n, is πi (t). Assuming that the trading of shares takes place continuously in a self-financing fashion, i.e., n X πi (t) dSi (t), dx(t) = S (t) i=0 i

then we have the following wealth equation  n o Pn   dx(t) = r(t)x(t) + b (t) − r(t) π (t) dt i i  i=1   Pm Pn + j=1 i=1 σij (t)πi (t)dW j (t),      x(0) = x .

(3)

(4)

0

The process π(·) := (π1 (·), · · · , πn (·))T is called a (self-financing) portfolio or trading strategy. Set B(t) := (b1 (t) − r(t), · · · , bn (t) − r(t))T , σ(t) := (σij (t))n×m .

(5)

With this notation, equation (4) becomes dx(t) = [r(t)x(t) + π(t)T B(t)]dt + π(t)T σ(t)dW (t), x(0) = x0 .

(6)

A standing assumption imposed throughout this paper is Standing Assumption. r ∈ L∞ F (0, T ; IR). Now we define an “allowable” portfolio. Definition 2.1. A self-financing portfolio π(·) is said to be admissible (on [0, T ]) n if it -valued, Ft -progressively measurable R t stochastic process, with R Tis an IR E 0 |π(s)T σ(s)|2 ds < +∞ and E supt∈[0,T ] | 0 π(s)T B(s)ds|2 < +∞.

Remark 2.1. As elaborated in the introduction section the definition of admissible portfolios is vital in studying a market. The class of admissible portfolios defined here is mathematically and economically general enough, yet not too (mathematically) general to bury the true economical essence or significance into sheer technicality. Specifically, in Definition R2.1 we work within the framework of square T integrability. The requirement that E 0 |π(s)T σ(s)|2 ds < +∞ is to make the corresponding stochastic integral in (6) R Twell defined and a martingale (we could weaken it, as in many existing works, to 0 |π(s)T σ(s)|2 ds < +∞ a.s. so that the integral is a local martingale; but then we would end up only with some routine technical complication – e.g. evoking a stopping timeR argument – rather than essential t difference). The requirement that E supt∈[0,T ] | 0 π(s)T B(s)ds|2 < +∞ (which is weaker than that π(·)T B(·) be square integrable) is a weak condition to ensure that the terminal wealth is a square integrable random variable, which is necessary

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in studying the contingent claim problem. Moreover, this condition ensures that any admissible portfolio stopped by an Ft -stopping time is still admissible. This is certainly motivated by the observation that investors may stop their investment according to a pre-determined stopping rule. Remark 2.2. When the investment opportunity set is deterministic and there exists a function θ(·) ∈ L2 (0, T, IRm ) such that σ(t)θ(t) = B(t), then it is easy to show, viaRthe Cauchy–Schwartz inequality, that a portfolio π(·) is admissible if and T only if E 0 |π(t)T σ(t)|2 dt < +∞.

Remark 2.3. The Standing Assumption that r ∈ L∞ F (0, T ; IR) is for notational RT convenience only. It can be weakened to 0 |r(s)|ds < +∞ a.s., in which case one only needs R t to modify the definition of an admissible R Tportfolio π(·) to such that E supt∈[0,T ] | 0 S0 (s)−1 π(s)T B(s)ds|2 < +∞ and E 0 |S0 (s)−1 π(s)T σ(s)|2 ds < +∞. Definition 2.2. An admissible portfolio π(·) is called an arbitrage opportunity on [0, t] ⊆ [0, T ] if there exists an initial x0 ≤ 0 and a time s ∈ [0, t], so that the corresponding wealth process x(·) satisfies P (x(s) ≥ 0) = 1 and P (x(s) > 0) > 0. Moreover, a market is called arbitrage-free on [0, t] if there exists no arbitrage opportunity on [0, t]. Definition 2.3. A European contingent claim ξ ∈ L2FT (IR) is said to be replicable if there exists an initial wealth x0 and an admissible portfolio π(·) such that the corresponding wealth process x(·) satisfies x(T ) = ξ. Moreover, a market is called complete (on [0, T ]) if any contingent claim ξ ∈ L2FT (IR) is replicable. Remark 2.4. In Ref. 13 (pp. 8–9, Example 2.3), an example is given where a continuous-time analogue of a doubling strategy is explicitly constructed to rationalize the usage of the tame portfolios. In the example, r(·) = 0, b(·) = 0 and σ(·) = 1. TheR constructed strategy is π(t) = (T − t)−1/2 1τ ≥t , where V t τ := inf{t ∈ [0, T ) : 0 (T − u)−1/2 dW (u) = α} T and α > 0 a given number. It is shown in Ref. 13 that π(·) is a self-financed portfolio with the corresponding wealth process, starting from zero wealth, satisfying x(T ) = α (hence an arbitrage opportunity), and π(·) is not a tame portfolio. However, such a portfolio is also ruled out by our Definition 2.1 even though we do not impose any uniform lower boundedness. Indeed, if π(·) is admissible according to Definition R t 2.1, then the corresponding wealth process with zero initial wealth is x(t) = 0 π(s)dW (s), which RT would be a martingale. This contradicts to the fact that 0 π(t)dW (t) = α > 0. 3. Some Preliminary Results In this section we present some preliminary results that are useful in proving the main results of this paper.

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Theorem 3.1. If the market is arbitrage-free on [0, T ], then there exists an F t progressively measurable stochastic process θ(·) such that σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s.. Proof: By a measurable selection argument [see Ref. 12 (Lemma A.3)] we can prove that there exists a progressively measurable process θ0 (·) such that θ0 (t) ∈ argminx∈IRm |σ(t)x − B(t)|2 . Define π0 (t) := σ(t)θ0 (t) − B(t). Then σ(t)T π0 (t) = 0 and π0 (t)T B(t) = π0 (t)T [σ(t)θ0 (t) − π0 (t)] = −|π0 (t)|2 . Define    −π0 (t)/|π0 (t)|2 if π0 (t) 6= 0, π1 (t) :=  0 if π0 (t) = 0. Then π1 (t)T σ(t) = 0 and π1 (t)T B(t) = 1π0 (t)6=0 (t) ≤ 1, implying that π1 (·) is an admissible portfolio. Let x(·) be the wealth process corresponding to π1 (·) with the initial wealth x(0) = 0. Then dx(t) = r(t)x(t)dt + π1 (t)T [B(t)dt + σ(t)dW (t)] = r(t)x(t)dt + 1π0 (t)6=0 (t)dt. Hence x(T ) =

Z

T

e 0

RT t

r(s)ds

1π0 (t)6=0 (t)dt ≥ emin{rT,0}

Z

T 0

1π0 (t)6=0 (t)dt ≥ 0, a.s.,

where r is a lower bound of the essentially bounded process r(·). If there is no Ft -progressively measurable stochastic process θ(·) satisfying σ(t)θ(t) = B(t), then RT P { 0 1π0 (t)6=0 (t)dt > 0} > 0. Thus, P {x(T ) > 0} ≥ P {

Z

T

0

1π0 (t)6=0 (t)dt > 0} > 0,

which suggests that π0 (·) is an arbitrage opportunity.

Q.E.D.

The converse of Theorem 3.1 requires stronger conditions on θ(·). Define ρ(t) := e−

Rt 0

|θ(s)|2 /2ds−

Rt 0

θ(s)T dW (s)

, t ∈ [0, T ].

(7)

TheoremR3.2. If there exists an Ft -progressively measurable stochastic process θ(·) T such that 0 |θ(s)|2 ds < +∞ a.s., σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s., and ρ(·) is a martingale with E|ρ(T )|2 < +∞, then the market is arbitrage free on [0, T ]. Moreover, in this case for any admissible portfolio π(·), the corresponding wealth RT process x(·) satisfies x(t) = ρ(t)−1 E(e− t r(s)ds x(T )ρ(T )|Ft ) ∀t ∈ [0, T ].

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Proof: For any admissible portfolio π(·), denote by x(·) the corresponding wealth process starting from 0. Then Itˆ o’s formula yields d[¯ x(t)ρ(t)] = [ρ(t)¯ π (t)T σ(t) − ρ(t)¯ x(t)θ(t)T ]dW (t), where x ¯(t) := x(t)/S0 (t) and π ¯ (t) := π(t)/S0 (t). Therefore x ¯(·)ρ(·) is a local martingale. On the other hand, Z t Z t |¯ x(t)| = | π ¯ (s)T B(s)ds + π ¯ (s)T σ(s)dW (s)| 0 0 Z t Z t ≤ sup | π ¯ (s)T B(s)ds| + sup | π ¯ (s)T σ(s)dW (s)| =: X. t∈[0,T ]

t∈[0,T ]

0

0

2

Then EX < +∞ by the admissibility of π(·). It follows then from the assumption on ρ(·) that |¯ x(t)ρ(t)| ≤ Xρ(t) ≤ X supt∈[0,T ] ρ(t) ∈ L1FT (IR), implying that x ¯(t)ρ(t) isRa martingale (via the dominated convergence theorem). Hence T x(t) = ρ(t)−1 E(e− t r(s)ds x(T )ρ(T )|Ft ), and the market is arbitrage free. Q.E.D. Recall that an equivalent martingale measure (on [0, T ]) is a probability measure Q so that 1) Q is equivalent to P , and 2) the discounted stock prices, {Si (t)/S0 (t) : t ∈ [0, T ]}, i = 1, · · · , n, are martingales under Q. Proposition 3.1. If there exists an arbitrage opportunity π(·) on [0, T ] with RT T 2 |π(t) σ(t)| dt < N a.s. for some N ∈ IR, then there is no equivalent martin0 gale measure on [0, T ] in the market. Proof: Suppose there is an equivalent martingale measure Q under which S¯i (t) := Si (t)/S0 (t), i = 1, R T· · · , n, are martingales on [0, T ]. Let π(·) be an arbitrage opportunity π(·) with 0 |π(t)T σ(t)|2 dt < N a.s. as assumed, and x(·) the corresponding wealth process with initial wealth 0. Then it follows from (3) that the discounted wealth process x ¯(·) := x(·)/S0 (·) satisfies n X πi (t) ¯ d¯ x(t) = dSi (t). S (t) i=1 i

Furthermore, for any t ∈ [0, T ], Z tX n X n πi (s) πj (s) =

0 i=1 j=1 Z tX n X n

Si (s) Sj (s)

S0 (s)−2 πi (s)πj (s)

0 i=1 j=1

≤

−2rT s−2 0 e

dhS¯i , S¯j i(s)ds

Z

n X

σik (s)σkj (s)ds

k=1 t

π(s)T σ(s)σ(s)0 π(s)ds 0

−2rT ≤ s−2 N. 0 e

It follows from the theory of stochastic integration with respect to a vector-valued local martingale [see, e.g., Ref. 11] that x ¯(·) is a local martingale under Q.

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On the other hand, d¯ x(t) = π ¯ (t)T B(t)dt + π ¯ (t)T σ(t)dW (t), x ¯(0) = 0, Rt where π ¯ (t) := π(t)/S0 (t). It follows that h¯ x, x ¯i(t) = 0 |¯ π (s)T σ(s)|2 ds; thus −2 −2rT Q Q N < +∞, where E is the expectation under probabilE h¯ x, x ¯i(T ) < s0 e ity Q. This shows that x ¯(·) is a martingale under Q [see, e.g., Ref. 18 (p. 129, Proposition 1.23)], which contradicts to π(·) being an arbitrage opportunity. Hence an equivalent martingale measure does not exist. Q.E.D. Corollary 3.1. If there exists an equivalent martingale measure on [0, T ] in the market, then there exists an Ft -progressively measurable stochastic process θ(·) such that σ(t)θ(t) = B(t), a.e. t ∈ [0, T ], a.s.. Proof: If the conclusion is not true, then by the proof of Theorem 3.1 there is an arbitrage opportunity π1 (·) with π1 (t)T σ(t) ≡ 0. Proposition 3.1 then applies to yield that there is no equivalent martingale measure. Q.E.D. 4. FTAP: Deterministic Investment Opportunity Set Theorem 3.2 gives a necessary condition for a general market with stochastic parameters to be arbitrage-free. To our best knowledge there is no equivalent condition available in literature for a continuous-time market with square integrable portfolios. However, if the investment opportunity set is deterministic, then we will derive elegant equivalent conditions including FTAP. Hence, throughout this section we impose the following assumption. Assumption (D) r(·), bi (·) and σij (·) are all deterministic. The key result of this paper follows. Theorem 4.1. Under Assumption (D), the market is arbitrage-free on [0, T ] if and only if there exists a function θ(·) ∈ L2 (0, T ; IRm ) such that B(t) = σ(t)θ(t) a.e. t ∈ [0, T ]. Before we give a rigorous proof of this result, let us sketch its main idea. The “if” part is fairly straightforward by applying Theorem 3.2. The “only if” part is much deeper, the proof of which is one of the main contributions of this paper. The basic idea is as follows. If the market is arbitrage-free, then by Theorem 3.1 there is a measurable function θ(·) satisfying σ(t)θ(t) = B(t). There might be many such functions; so we take the “minimal” one by setting θ0 (t) := argminσ(t)θ=B(t) |θ|2 . Our target is then to prove that θ0 (·) ∈ L2 (0, T ; IRm ). We will prove it by connamely, we need to construct an arbitrage opportunity π(·) whenever Rtradiction, T 2 |θ (t)| dt = +∞. 0 0 We complete the construction (precisely speaking, the construction is for an auxiliary, simpler version of the wealth equation (6), see (9) below, which is nonetheless sufficient) for three different cases: Case 1: There exists a sub-interval [a, b] ⊆ [0, T ]

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Rb Rb such that a |θ0 (s)|2 ds = +∞ and t |θ0 (s)|2 ds < +∞ for any t ∈ (a, b). In other RT words, the infinity of the integration 0 |θ0 (t)|2 dt = +∞ is due to the ill-behavior, so to speak, of θ0 (·) in the right neighborhood R b of a certain pint a; Case R t 2: There exists a sub-interval [a, b] ⊆ [0, T ] such that a |θ0 (s)|2 ds = +∞ and a |θ0 (s)|2 ds < +∞ for any t ∈ (a, b). In this case, θ0 (·) is ill behaved in the left neighborhood of a cerRT tain pint b; and Case 3: Neither (i) nor (ii) occurs but overall 0 |θ0 (t)|2 dt = +∞. Notice that the last case may indeed happen. It turns out the constructions of arbitrage opportunities (bear in mind that the constructed portfolios must be admissible per Definition 2.1) for these three cases are very different, and all delicate. Yet, all the constructions are based on the following lemma of functional analysis. Lemma 4.1. A measurable function f ∈ L2 (0, T ; IRm ) if and only R T if 2 RT m T 2 | 0 f (t) g(t)dt| < +∞ for any g ∈ L (0, T ; IR ). Moreover, if 0 |f (t)| dt = +∞, then there exists g(·) ∈ L2 (0, T ; IRm ), which can be represented as g(·) R T = k(·)f (·) for a measurable function k(·) with k(t) ≥ 0 a.e. t ∈ [0, T ], such that 0 f (t)T g(t)dt = +∞. To prove this lemma we first need the following technical result. Lemma 4.2. Let {an : n = 1, 2, · · · } be a nonnegative sequence with a1 > 0 and P∞ Pn n=1 an = +∞, and let bn := i=1 ai . Then ∞ X an = +∞, b n=1 n

Proof: Observe

ai bi−1

=

bi bi−1

and

∞ X an < +∞. b2 n=1 n

(8)

bi − 1 ≥ ln bi−1 for any integer i ≥ 2. Hence

∞ ∞ X X ai ≥ (ln bi − ln bi−1 ) = +∞, ∀n ≥ 2. b i=n i−1 i=n

Now, if for any integer N > 0 there exists n > N such that abnn > 12 , then clearly P ∞ an n=1 bn = +∞. Therefore we may assume that there exists an integer N > 0 such that for any n > N , abnn ≤ 12 . In this case, for any n > N , bn−1 ≥ 12 bn , implying an an bn ≥ 2bn−1 and, thus, ∞ ∞ X an 1 X an ≥ = +∞. bn 2 bn−1

n=N +1

n=N +1

This proves the first equality of (8). For the second inequality of (8), notice that

Thus,

P∞

n=1

an bn − bn−1 bn − bn−1 1 1 = ≤ = − , ∀n ≥ 2. b2n b2n bn bn−1 bn−1 bn P∞ an 1 1 1 2 n=2 ( bn−1 − bn ) = a1 < +∞. b2 ≤ a1 + n

Q.E.D.

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Proof of Lemma 4.1. The if ” part is obvious. To prove the “if” part, it R T “only 2 suffices to show that if 0 |f (t)| dt = +∞ with a measurable function f on [0, T ], RT then there exists g(·) ∈ L2 (0, T ; IRm ) such that 0 f (t)T g(t)dt = +∞. To this end, for any integer n ≥ 1, denote In := {t ∈ [0, T ] : |f (t)| ∈ [n − 1, n)}, I∞ := {t ∈ [0, T ] : |f (t)| = +∞}, and d(t) := min{n ∈ N : |f (t)| < n}. Clearly d(t) = n if t ∈ In . On the other hand, if µ(I∞ ) > 0, where µ is the Lebesgue measure, then g := |ff | 1f 6=0 is the desired function. So we assume hereafter that µ(I∞ ) = 0. R Define an := In |f (t)|2 dt, n = 1, 2, · · · . Without loss of generality, suppose Pn f (t) a1 > 0. Define bn := i=1 ai and g(t) := bd(t) , t ∈ [0, T ]. Then we have the following identities Z T Z T ∞ Z ∞ Z ∞ X X X |f (t)|2 |f (t)|2 an |f (t)|2 dt = dt = dt = , |g(t)|2 dt = 2 2 2 bd(t) bd(t) bn b2 0 0 n=1 In n=1 In n=1 n and Z T

Z

∞ Z ∞ Z ∞ X X X |f (t)|2 |f (t)|2 |f (t)|2 an dt = dt = dt = . b b b b n d(t) d(t) 0 0 n=1 In n=1 In n=1 n RT RT P∞ Since 0 |f (t)|2 dt ≡ n=1 an = +∞, Lemma 4.2 applies to yield 0 |g(t)|2 dt < +∞ RT and 0 f (t)T g(t)dt = +∞. Finally, the second assertion of the lemma is evident from the constructive proof above. Q.E.D.

f (t)T g(t)dt =

T

Remark 4.1. Lemma 4.1 appears to be a standard result in functional analysis. Yet, surprisingly, we have searched this result in major texts in functional analysis to no avail. Hence we supply a proof here. Remark 4.2. Lemma 4.1 is valid if the interval [0, T ] is replaced by any [a, b] ⊆ [0, T ]. Indeed, based on its proof the lemma can be extended to any measure space (S, G, µ) with µ(S) < +∞. Next let us consider the following auxiliary, “controlled” Itˆ o process dx(t) = u(t)T θ(t)dt + u(t)T dW (t), x(0) = 0,

(9)

where θ(·) is a given IRm -valued measurable function with |θ(t)| < +∞ a.e. t ∈ [0, T ], and u(·) ∈ L2F (0, T ; IRm ) can be regarded as a control. Rb Lemma 4.3. If there is an interval [a, b] ⊆ [0, T ] such that a |θ(t)|2 dt = +∞ and Rb |θ(t)|2 dt < +∞ for any t ∈ (a, b), then there exists u(·) ∈ L2F (0, T ; IRm ) with RtT |u(t)|2 dt < N, a.s. for some N ∈ IR, u(t)T θ(t) ≥ 0, a.s., a.e. t ∈ [0, T ] and 0 Rt E supt∈[0,T ] | 0 u(s)T θ(s)ds|2 < +∞, so that the corresponding Itˆ o process in (9) satisfies x(T ) ≥ 0 a.s., and P (x(T ) > 0) > 0. Moreover, u(t) = α(t)θ(t) for some Ft -progressively measurable process α(·) ≥ 0.

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Proof: By Lemma 4.1, there exists a nonnegative measurable function k(·) on m 2 T 2 [a, b] such R b thatT g(·) := k(·)θ(·) satisfies g(·) ∈ L (a, b; IR ), g(·) θ(·) ≡ k(·)|θ(·)| ≥ 0, and a g(t) θ(t)dt = +∞. Extend k(·) to [0, T ] (still denoted by k(·)) by setting zero value outside of [a, b]R and let g(·) := k(·)θ(·). T Define f (t) := exp{− t g(s)T θ(s)ds}. Then by the assumption on θ(·) one has f = 0 on [0, a], f > 0 on (a, b], f = 1 on (b, T ], and f 0 (t) = f (t)g(t)T θ(t) ≥ 0 ∀t ∈ (0, T ] (hence f Ris non-decreasing). V t / [ 41 , 1]} b, and u(t) := Set y(t) := 21 + 0 g(s)T dW (s), τ := inf{t ≥ 0 : y(t) ∈ RT RT f (t)g(t)1t a) > 0, and 0 |u(t)|2 dt ≤ 0 |f (t)g(t)|2 dt ≤ RT |g(t)|2 dt < +∞. Moreover, u(t)T θ(t) = f (t)k(t)|θ(t)|2 1t a) > 0). Finally, u(t) = α(t)θ(t) with α(t) := k(t)f (t)1t a a.s.. On the other hand, τ < b a.s. by Rb Rτ virtue of the fact that 0 g(t)T θ(t)dt = +∞. As a consequence, 0 g(s)T θ(s)ds−1 = Rτ RT RT − 0 g(s)T dW (s). Now define u(t) := g(t)1t≤τ . Then 0 |u(t)|2 dt ≤ 0 |g(t)|2 dt
0) > 0. Moreover, u(t) = α(t)θ(t) for some Ft -progressively measurable process α(·) ≥ 0. Proof: If the conclusion is not true, then we claim the following Z a1 Z T There exists a1 ∈ (0, T ) such that |θ(t)|2 dt = +∞, |θ(t)|2 dt = +∞. (11) a1

0

R a0 Indeed, if (11) is false, then take b := inf{a0 ≥ 0 : 0 R|θ(t)|2 dt = +∞}. We a must have 0 < b < T (if b = 0, then for any a ∈ (0, T ), 0 |θ(t)|2 dt = +∞, and RT hence a |θ(t)|2 dt < +∞ since (11) is not true. This, together with the fact that RT 2 0 |θ(t)| dt = +∞, would lead to the conclusion of the theorem in view of Lemma 4.3. On the other hand, if b = T , then the conclusion of the theorem is valid by virtue of Lemma 4.4). Now, to invalidate (11) we must have either Z b Z T |θ(t)|2 dt = +∞ and |θ(t)|2 dt < +∞. (12) b

0

or Z

b 2

|θ(t)| dt < +∞ and 0

Z

T

|θ(t)|2 dt = +∞.

(13)

b

Ra First suppose (12) is the case. By the definition of b we have 0 |θ(t)|2 dt < +∞ for any a ∈ (0, b). Hence it follows from Lemma 4.4 that the conclusion of the theorem holds true which by the R a is a contradiction. Next, if (13) is true. Then Ragain T definition of b, 0 |θ(t)|2 dt = +∞ for any a ∈ (b, T ). Consequently, a |θ(t)|2 dt < +∞ for any a ∈ (b, T ), for otherwise (11) would be true. However, this along with the second equality of (13) yields that the conclusion of the theorem is valid due to Lemma 4.3. We have now proved and we can apply trick to [a1 , T ] to find an R a(11), R Tthe same 2 2 2 a2 ∈ (a1 , T ) such that a1 |θ(t)| dt = +∞ and a2 |θ(t)| dt = +∞. Continuing with this, find a strictly increasing sequence {ai , i = 1, 2, · · · } on (0, T ) such that R ai we can 2 |θ(t)| dt = +∞. Denote c := limi→+∞ ai > a1 . ai−1 On any interval (ai , ai+1 ), it follows from Lemma 4.1 (and Remark 4.2) that there a measurable k(·) ≥ 0 such that g(·) = k(·)θ(·) ∈ L2 (aRi , ai+1 ; IRm ) and R ai+1 exists a g(t)T θ(t)dt = +∞. By scaling properly we can assume that aii+1 |g(t)|2 dt ≤ ai

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2−i . On the other hand, define    g(t), if g(t)T θ(t) ≤ k, gk (t) :=   0, if g(t)T θ(t) > k,

k = 1, 2, · · · .

Evidently gk (t) → g(t), a.e. t ∈ [0, T ] as k → +∞. Noticing Rthat each gk (t)T θ(t) a is nonnegative, we apply Fatou’s lemma to obtain limk→+∞ aii+1 gk (t)T θ(t)dt ≥ R ai+1 g(t)T θ(t)dt = +∞. In particular, we can take k sufficiently large to ensure ai R a that aii+1 gk (t)T θ(t)dt > 1. To summarize, we have proved that on (ai , ai+1 ) there exists a measurable function, denoted by gi (·), with the following properties: Z ai+1 Z ai+1 2 −i |gi (t)| dt ≤ 2 , 1 < gi (t)T θ(t)dt < +∞. (14) ai

ai

Rt RT Denote h(t) := i=1 gi (t)1t∈(ai ,ai+1 ] . Then 0 |h(t)|2 dt ≤ 1, 0 h(s)T θ(s)ds < Rt +∞ for any t < c, and meanwhile 0 h(s)T θ(s)ds−→+∞ as t → c. Set τ := inf{t ≥ Rt Rt 0 : 0 h(s)T θ(s)dt − 1 ≥ − 0 h(s)T dW (s)}. The preceding properties of h(·) imply that 0 < τ < c, a.s.. Hence employing exactly the same approach as in the proof of Lemma 4.4 we conclude that u(t) := h(t)1t 0) > 0. Moreover, RT t |u(t)|2 dt < N, a.s. for some N ∈ IR, and E supt∈[0,T ] | 0 u(s)T θ0 (s)ds| < +∞. 0 Define π(t) := α(t)λ(t). Then π(t)T σ(t) = α(t)λ(t)T σ(t) = α(t)θ0 (t)T = u(t)T ,

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and π(t)T B(t) = α(t)λ(t)T B(t) = α(t)λ(t)T σ(t)θ0 (t) = u(t)T θ0 (t). Together with (15) this means that π(·), being an admissible portfolio, is indeed an arbitrage opportunity, which leads to a contradiction. Finally, if r(·) 6≡ 0, then (¯ x(·), π ¯ (·)) := (x(·)/S0 (·), π(·)/S0 (·)) satisfies a wealth equation with r(·) = 0. In view of the Standing Assumption that r(·) is essentially bounded, the result follows readily. Q.E.D. We finally arrive at our FTAP. Theorem 4.2. Under Assumption (D), the following three statements are equivalent. (a) The market is arbitrage free on [0, T ]. (b) There is an equivalent martingale measure on [0, T ]. (c) There exists θ(·) ∈ L2 (0, T ; IRm ) such that B(t) = σ(t)θ(t) a.e. t ∈ [0, T ]. Proof: Theorem 4.1 shows that that (a) and (c) are equivalent. Now, if (c) holds, then define a new probability measure Q by Rt Rt 2 T dQ := e− 0 |θ(s)| /2ds− 0 θ(s) dW (s) , t ∈ [0, T ]. dP Ft

Girsanov’s theorem along with Novikov’s condition yields that Q is an equivalent martingale measure; hence (b). Conversely, suppose (b) holds. By Corollary 3.1, there exists θ(·) such that σ(t)θ(t) = B(t) a.e. t ∈ [0, T ]. Defining θ0 (·) := argminσ(t)θ=B(t) |θ|2 , we claim that θ0 (·) ∈ L2 (0, T ; IRm ). Indeed, if this is not true, then employing an exactly the same argument as that in the proof of Theorem R T 4.1 we conclude that there exists an arbitrage opportunity π(·) on [0, T ] with 0 |π(t)T σ(t)|2 dt < N, a.s. for some N ∈ IR. It follows from Proposition 3.1 then that there exists no equivalent martingale measure in the market, which is a contradiction. This proves (c). Q.E.D. Theorem 4.2 (especially the equivalence between (a) and (c)) provides very convenient and easy way to check whether or not a market (with a deterministic investment opportunity set) is arbitrage free. Here are a couple of examples.

Example 4.1. Consider a market where there is a bond and a stock. Suppose r(t) = 0, b(t) = 0.1, σ(t) = 0.1tα (α > 0), and T = 1 (year). For any α > 0, the volatility rate of the stock starts off very small, and gradually increases and reaches maximum at the terminal time. Now, the only θ(·) (up to the exception on a null-set) satisfying σ(t)θ(t) = B(t) is θ(t) = t−α (t > 0). Thus according to Theorem 4.2 the market is arbitrage-free if and only if θ(·) is square integrable, or α ∈ (0, 12 ). This shows that the variability of the market crucially depends on the value of α and, in particular, α = 12 is the critical number.

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Example 4.2. For a market with multiple stocks and time-invariant market coefficients, the arbitrage-free condition boils down to the solvability of the algebraic equation σθ = B (with θ being the unknown). There are of course many convenient algebraic criteria to check the latter. 5. Counterexamples: Stochastic Investment Opportunity Set Given the results in the previous section, it is tempting to extend them to the case with a stochastic investment opportunity set. Unfortunately, this cannot be done. In this section, we show, via two counterexamples, that with a stochastic investment opportunity set Theorem 4.1 is no longer true in either directions if we replace θ ∈ L2 (0, T, IRm ) by θ ∈ L2F (0, T, IRm ). Example 5.1. In this example we show that arbitrage-free does not ensure the existence of θ ∈ L2F (0, T, IRm ) satisfying B(t) = σ(t)θ(t). Consider a market with one bond and one stock, with r(t) = 0, B(t) = 1, and σ(t) = 1t 0) > 0. Since the market coefficients are deterministic constants up to t0 , there is no arbitrage opportunity on [0, t0 ]; so t¯ > t0 . On the other hand, for almost all ω, the following equation Z t Z t 1 x(t) = x(t0 , ω) + π(s)ds + π(s)dW (s), t ∈ (t0 , T ] ξ(ω) t0 t0 can be regarded as a wealth equation associated with a market, which has deterministic coefficients, on the probability space (Ω, F, P (·|Ft0 )(ω)) and a time horizon [t0 , T ]. So we can apply Theorem 3.2 to obtain x(t0 ) = E(x(t¯)e−ξ

2

(t¯−t0 )/2−ξ(W (t¯)−W (t0 ))

|Ft0 ) a.s..

It follows that P (x(t0 ) ≥ 0) = 1, P (x(t0 ) > 0) > 0, which is impossible because there is no arbitrage opportunity on [0, t0 ]. Example 5.2. This example demonstrates that the existence of θ ∈ L2F (0, T, IRm ) satisfying B(t) = σ(t)θ(t) needs not lead to the absence of arbitrage. Consider a 5dimensional Bessel process R(·) starting from 1 (namely, R(·) is the radial part of a 5-dimensional Brownian motion with R(0) = 1), z(t) := R(t)2 , and v(t) := R(t)−1 . By the theory of Bessel process [see, e.g., Ref. 18, p. 439], p ¯ (t) (16) dz(t) = 5dt + 2 z(t)dW

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¯ (·). On the other hand, inf 0≤t 0, such that ξ T σt σtT ξ ≥ c0 |ξ|2 ,

∀ξ ∈ IRn , t ∈ [0, T ].

(H2) The death benefit function g : [0, T ] × IRd × IR 7→ IR+ is bounded, and continuously differentiable, with bounded derivatives. 3. The HJB Equations Let us begin by considering the simplest UVL insurance models, in which the only uncertainty for the termination (or decrement) is death, derive the corresponding HJB equations via two different pricing strategies, depending on the structure of the death benefits. To simplify presentations, we shall assume d = 1 throughout the section. However, we should note that all the analysis can be generalized to higher dimensional cases without substantial difficulties. Since the benefit is paid at fixed time, we will drop T in the benefit function g. To begin with, let us consider an intermediate stochastic control problem with the cost functional: 4 ˜ w, s, z; π) = J(t, Et,w,s,z {u(WTπ − g(ST , ZT ))}.

(14)

This can be thought of as that the death has occurred before T , hence XT ≡ 1. The utility maximization problem is then reduced to a standard stochastic control problem. Denoting the value function as ˜ w, s, z; π), ˜ (t, w, s, z) = sup J(t, U

(15)

π∈A

˜ is at least the unique viscosity solution to the following then it is well-known that U the HJB equation (cf, Ref. 17):  2 1 2 2˜ Z  ˜ ˜ ˜ ˜  0 = Ut + max σ π Uww + π σ sUws + σ zσ Uwz + (µ − r)Uw ]   π 2     1 2  ˜ss + 1 (˜ ˜zz + σσ Z sz U ˜sz σ 2 + σ Z )z 2 U + σ 2 s2 U (16) 2 2   Z  ˜s + µ z U ˜z ;  +rwU˜w + µsU     ˜ U (w, T, s, z) = u(w − g(s, z)). To derive the HJB equation for the original optimization problem, we first argue heuristically. By virtue of the Bellman Principle (of dynamic programming, cf.

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Ref. 17), and using the total probability formula we have, for any h > 0 and any admissible portfolio π ∈ A, π U (w, t, s, z) ≥ Et,x,s,z {U (t + h, Wt+h , St+h , Zt+h )} π = Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )|T (x) > t + h}P {T (x) > t + h} π +Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )|T (x) ≤ t + h}P {T (x) ≤ t + h} π = Et,w,s,z {U (Wt+h , t + h, St+h , Zt+h )}h px+t π ˜ +Et,w,s,z {U(W , t + h, St+h , Zt+h )}h qx+t .

(17)

t+h

Here we used the fact that the future-life-time random variable T (x) is independent of the processes (W π , S, Z); and that, given T (x) ≤ t + h, the optimization problem (11) on [t + h, T ] is the same as (15). We remark that the argument above actually reflect a pricing strategy: when death occurs before t + h became known, then one should simply carry out the optimization problem knowing that the terminal wealth will be deducted by the amount g(ST , ZT ) at time T . Such a strategy is quite generic, hence particularly reasonable when the non-tradable assets are present, since there seems to no simple way to hedge the risk g(ST , ZT ) a priori. ˜ are smooth. Applying Itˆ Suppose now that both value functions U and U o’s ˜ (t, Wt , St , Zt ), respectively, and noting that h px+t + formula to U (t, Wt , St , Zt ) and U h qx+t = 1, one shows that (suppressing variables) ˜ w, s, z)]h qx+t [U (t, w, s, z) − U(t, Z t+h 1 ≥ Et,w,s,z { (Ut + (rW + π(µ − r)Uw ) + sUs µ + zUz µZ + σ 2 (Uww π 2 + s2 Uss ) 2 t 1 2 2 + z Uzz (˜ σ 2 + σ Z ) + sUws πσ 2 + zUwz πσσ Z + zsUsz σσ Z )du}h px+t (18) 2 Z t+h +Et,w,s,z { ...du}h qx+t . t

In the above the integrand of the last integral is similar to that of the first one ˜ Now we divide both sides of (18) by h and let h → 0. with U being replaced by U. Noting that 0 px+t = 1 and h qx+t /h → λx (t), we obtain the following HJB equation for U :  1   [(µ − r)πUw + σ 2 π 2 Uww + (Uws Sσ 2 + Uwz Zσ Z σ)π]   0 = Ut + max π 2     1 1 2 +rwUw + sUs µ + zUz µZ + σ 2 s2 Uss + z 2 Uzz (˜ σ2 + σZ ) (19) 2 2      +szUsz σσ Z + λx (t)(U˜ − U )    U (T, w, s, z) = u(w),

˜ satisfies (16). where U

The Special Case: g(s, z) ≡ g(s).

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First note that in this case the non-tradable asset Z does not appear in the previous argument, thus (19) is reduced to  1 1   0 = Ut + max[(µ − r)πUw + σ 2 π 2 Uww + (σ 2 sUws )π] + σ 2 s2 Uss   π 2 2 (20) ˜ +rwU + sµU + λ (t)( U − U ) = 0,  w s x    U (T, w, s) = u(w), and (16) becomes:  ˜ww + (σ 2 sU ˜ws )π] + 1 s2 σ 2 U ˜ss ˜t + max[(µ − r)π U˜w + 1 σ 2 π 2 U  0 = U π 2 2 ˜s +rwU˜w + µsU  ˜ U (T, w, s) = u(w − g(s))

(21)

We should point out that in this case one can actually use a more specific strategy if the market is complete. In fact, considering the benefit payment as a contingent claim, one can easily identify its current market price with which the payment amount g(ST ) can be replicated at time T . Simply setting aside the current price of the benefit payment, one can then proceed the optimization problem as if there is no insurance risk involved at all. Such a strategy was used in Ref. 18 in the case g ≡ 1. In what follows we give a brief sketch of the argument to modify the result to our case, and we show that the two strategies will actually produce the same result. First recall value function V 0 defined by (9). Assuming that all the market parameters r, µ, and σ are deterministic, continuous functions then it is well-known that the value function V 0 is C 1,2 ([0, T ) × IRd ), and it satisfies the following HJB equation: (cf. e.g., 18 ): ( 1 0 0 = Vt0 + max{ |σ T π|2 Vww + π(µ − r)Vw0 } + rwVw0 , (22) π 2 V 0 (T, w) = u(w). Next, we consider the value function V , defined by (10). Applying the Bellman Principle and the total probability formula again, we can show that a counterpart of (17) holds: V (t, w, s) ≥ Et,w,s {V (t + h, Wt+h , St+h )}h px+t +Et,w,s {V 0 (t + h, Wt+h − c(t + h, St+h ))}h qx+t ,

(23)

where c(t, s) is the market price of the contingent claim g(ST ). We note that in the above we deducted c(t + h, St+h ) from the wealth at t + h and then carried out the “future” optimization problem without the insurance risk. Thus the value function ˜ in (17)!). becomes V 0 (t + h, Wt+h − c(t + h, St+h )) (compared to U Now repeating the same argument as before, assuming that V and V 0 and c are all smooth, and using the fact that c(·, ·) satisfies the Black-Scholes PDE, it is fairly

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easily checked that V satisfies the following HJB equation (suppressing all variables for V ):  1   0 = Vt + max{ σ 2 π 2 Vww + (µ − r)πVw + σ 2 πsVws } + rwVw + µsVs π 2 (24) + 21 σ 2 s2 Vss + λx (t)(V 0 (t, w − c(t, s)) − V )   V (T, w, s) = u(w)

At this point it should be clear that this strategy will not work in general since the arbitrage price c(t, s) is not uniquely determined if the payoff contains the nontradable asset ZT ! One should also note that the two different strategies yield almost the same HJB equations (20) and (24), the only difference is that the last ˜ (t, w, s) in (20) is replaced by V 0 (t, w − c(t, s)) in (24). The following result term U clarifies this point even further. Theorem 3.1. Assume (H1), and assume that the benefit function g ≡ g(s). Then, it holds that V (t, w, s) ≡ U (t, w, s), for all (t, w, s) ∈ [0, T ] × IR+ × IR+ .

Proof. Comparing (24) and (19), it is clear that we need only verify that ˜ (t, w, s) = V 0 (t, w − c(t, s)), ∀(t, w, s). U To this end, let us first recall from (5) that for the given initial state w and the portfolio π, the wealth process satisfies the SDE Z t Z t Wt = w + (ru Wu + hπu , µu − ru 1i)du + hπu , σu dBu i, (25) 0

0

w,π

and we denote the solution by W to specify the dependence on w and π. It is by now well-known (see, e.g., 8 ) that under (H1) the Black-Scholes price c(·, ·) for the contingent claim g(ST ) satisfies the Black-Scholes PDE:  ∂c 1 2 2 ∂ 2 c  ∂c + rt s + σ s − rt c(t, s) = 0, (26) ∂t ∂s 2 t ∂s2  c(T, s) = g(s), 4

and the process Yt = c(t, St ), t ≥ 0, can be expressed as the unique solution to the following “backward stochastic differential equation” (BSDE): dYt = rt Yt dt + hπt0 , µt − rt 1idt + hπt0 , σt dBt i, (27) YT = g(ST ), where π 0 ∈ A is the “hedging strategy” of the claim g(ST ). 4

Let us now define a mapping T : A 7→ A by T (π) = π 0 = π + π 0 , π ∈ A. Then clearly T is one-to-one mapping, so that T (A) = A. Furthermore, the linearity of 0 the equation (25) and the uniqueness of the solution to SDE implies that Wtw,π = w−c(0,s),π Yt + W t , for all t ∈ [0, T ] . In particular, at terminal time T , this becomes w−c(0,s),π

WT

0

= WTw,π − g(ST ),

∀π ∈ A.

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˜ we have Finally, by the definition of the value functions V 0 and U w−c(0,s),π

V 0 (0, w − c(0, s)) = sup E0,w {u(WT

)}

π∈A

w,T (π)

= sup E0,w {u(WT π∈A

− g(ST ))}

0

˜ (0, w, s), = sup E0,w {u(WTw,π − g(ST ))} = U π 0 ∈A

proving the theorem. 4. The Case of Exponential Utility In this section we consider a special type of utility function—the “exponential utility”. Such a utility function has been widely used in practice, especially in actuarial mathematics. In fact, the premium principle given by (13) has been known to have certain very desirable properties if and only if the utility function is exponential (cf. e.g., Ref. 4). We should note that while the discussion in this section could be considered as a generalized version of Example 4.1 of Ref. 18, the generality of our benefit function g makes the problem a little more involved. Let us be more specific. In what follows we shall assume that the utility function takes the form 1 4 u(w) = − e−αw , w ∈ IR. (28) α Then, recall from 18 that in this particular case the HJB equation (22) has the following explicit solution. (µ − r)2 1 exp{−αwer(T −t) − (T − t)} (29) α 2σ 2 Our discussion will depend heavily on some classical results in non-linear PDEs. In particular, our solution will not be possible if certain reaction-diffusion equation does not have global solution. But fortunately all the PDEs involved in our discussion can be shown to be well-posed, therefore the solution process can be carried out to the end. We shall discuss two cases of benefit functions separately. V 0 (t, w) = −

A. The case g = g(s). As we mentioned before, in this case the contingent claim g(ST ) is hedgeable, and its price c(·, ·) satisfies the Black-Scholes PDE (26), the optimal price of the UVL is relatively easier to obtain. We have the following theorem. Theorem 4.1. Assume (H1), and assume that utility function u takes the form (28). Suppose also that the benefit function g ≡ g(s), s ≥ 0, and that the force of mortality λx (t), t ≥ 0, are both bounded and deterministic. Then, the solution to (24) can be written as V (t, w, s) = V 0 (t, w) exp{αc(t, s)er(T −t) − h(t, s)},

(30)

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where c(·, ·) is the classical solution to the Black-Scholes equation (26), and h(·, ·) is the classical solution to the following reaction-diffusion system: ( 1 0 = ht + srhs + σ 2 s2 hss − λx (t)(eh − 1) (31) 2 h(T, s) = αg(s) Furthermore, the optimal premium of the UVL insurance is given by p = c(0, s) −

h(0, s) −rT e . α

(32)

Proof. First note that the solution to the HJB equation (24) is unique (at least in the viscosity sense), we need only show that a classical solution to (24) exists. To this end, recall that the value function V 0 (t, w), given explicitly by (29), is a smooth function, concave in w, and it satisfies the PDE:  2 (µ − r)2 Vw0  + rwVw0 0 = Vt0 − 2V 0 (33) 2σ ww  0 V (T, w, s) = u(w).

We shall look for a classical solution of (24) with the special form: V (t, w, s) = V (t, w)Φ(t, s). Note that any solution of such a form will be concave (and C 2 ) in w, thus we can solve the maximum in (24) by choosing 0

π∗ = −

(µ − r)Vw + σ 2 sVws , σ 2 Vww

and the equation (24) becomes  ((µ − r)Vw + σ 2 sVws )2 1   0 = V − + rwVw + µsVs + σ 2 s2 Vss  t  2 2σ Vww 2 0 +λ (V (t, w − c(t, s)) − V )  x    V (T, w, s) = u(w).

(34)

Plugging in V = V 0 Φ we obtained from (34) that

2 n o (µ − r)2 Vw0 1 0 = Vt0 − + rwVw0 + V 0 (Φt + sµΦs + σ 2 s2 Φss ) 2 0 2σ Vww 2 2 V0 s2 σ 2 Φ2s +λ(t) V 0 (t, w − c(t, s)) − V 0 Φ) − w0 ((µ − r)sΦs − 0 Φ Vww 2Vww

In the above, the first {· · · } vanishes because of (33). Also, using the explicit form (29) of V 0 , and with some straightforward computation we deduce that Φ must satisfy the following PDE:  1 Φ2  Φt + rsΦs + σ 2 s2 Φss − s + λx (t) exp{c(t, s)αer(T −t) } − Φ = 0, (35) 2 Φ  Φ(T, s) = 1.

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We now show that the equation (35) has a classical solution. To see this, we first consider the transformation: h(t, s) = c(t, s)αer(T −t) − ln Φ(t, s), (t, s) ∈ [0, T ] × IR. Then it is readily seen that (suppressing variables): ht = ct αer(T −t) − cαrer(T −r) −

Φt ; Φ

hs = cs αer(T −t) −

Φs ; Φ

Φss (Φs )2 + . Φ Φ2 Using these relations one easily verifies that h satisfies the PDE:  1   0 = αer(T −t) (ct + rscs + σ 2 s2 css − rc) − ht − srhs 2 − 21 σ 2 s2 hss + λx (t)(eh − 1)   h(T, s) = αc(T, s) hss = css αer(T −t) −

(36)

Since c(·, ·) solves the Black-Scholes PDE (26), (36) becomes (31). Furthermore, if we make a change of variables: v = log s and τ = T − t in (31), and denote b h(τ, v) = h(T − τ, ev ), then b h satisfies the following “reaction-diffusion” equation: ( 1 1 hτ − (r − σ 2 )hv − σ 2 hvv + λx (T − τ )(eh − 1) = 0 (37) 2 2 h(0, v) = αg(ev ) We should note that the equation (37) is a semilinear parabolic PDE with exponential growth, which in general may have a finite time blow-up. But by examing the particular form of the nonlinear term carefully and applying the results in Ladyzenskaja7 we can show the equation (37) does have a bounded classical solu4 tion b h (see 9 for details). Therefore h(t, s) = b h(T − t, log s) is a solution to (31), and

V (t, w, s) = V 0 (t, w)Φ(t, s) = V 0 (t, w) exp{αc(t, s)er(T −t) − h(t, s)} is a classical solution to (24), proving the first part of the theorem. To conclude the proof, we recall that by the Principle of Equivalent Utility, the optimal premium is defined by: V 0 (0, w) = V (0, w + p, s). By virtue of Theorem 4.1, this relation becomes: V 0 (0, w) = V 0 (0, w + p) exp{αc(0, s)erT − h(0, s)}.

(38)

Now the conclusion follows from the explicit form (29) of V 0 (0, w), and some fairly simple calculations. We leave the details to the interested reader, and the proof is now complete. We remark that if the force of mortality λx (t) ≡ 0, that is, the death never occurs, then it is easily checked that h(t, s) = αc(t, s)er(T −t) satisfies (31), thus p ≡ 0 by (32), as it should be. B. The case g = g(s, z).

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In this case the explicit formula for the price of UVL price is a little more complicated, due to the presence of the non-tradable asset Z in the death benefit (hence the contingent claim g(ST , ZT ) is no longer hedgable in general). But we can still proceed along the same line as before, and possibly with slightly different notations, we have the the following theorem. Theorem 4.2. Assume that the utility function is of the form u(w) = − α1 e−αw . Assume also that the benefit function g(·, ·) and the force of mortality λx (·) are both smooth and bounded. Then the optimal premium can be written as p(t, s, z) =

1 −r(T −t) e h(T − t, log s, log z), α

(39)

where h is a bounded, classical solution to the PDE  1 2 2 1 1 2  ˜ hy2 − σ 2 hy1 y1 − (˜ σ + σ z 2 )hy2 y2 − σσ z hy1 y2  hτ − σ   2 2 2    1 ˜2 + σz 2  µ−r z σ − r − σ 2 h y1 − µ z − σ − h y2 2 σ 2   ˜   −λx (T − τ )(eh−h − 1) = 0;     h(0, y1 , y2 ) = 0,

˜ is a bounded, classical solution to the PDE: and h  z˜ 2 z2 ˜ ˜ 2 − 1 σ2 h ˜ y y − 1 (˜ ˜  h − 1σ ˜2 h y2 1 1  2 2 σ + σ )hy2 y2 − σσ hy1 y2  τ 2  1 2 ˜ µ−r z σ ˜ 2 + σ z 2 ˜ z − r − hy2 = 0; σ h − µ − σ − y 1  2 σ 2   ˜ h(0, y1 , y2 ) = αg(ey1 , ey2 )

(40)

(41)

Proof. The idea of the proof is similar to that of Theorem 4.1. But this time we seek solutions of (16) and (19) with u(w) = − α1 e−αw such that they are of the special forms ˜ −t,ln s,ln z) ˜ (t, w, s, z) = V 0 (t, w)eh(T U ,

and U (t, w, s, z) = V 0 (t, w)eh(T −t,ln s,ln z) ,

respectively. Following the same arguments as in Theorem 4.1, one shows that ˜ s, z) and h(τ, s, z) will have to solve the reaction-diffusion equations (41) and h(τ, (40), respectively, after a change of variables τ = T − t, y1 = ln s, and y2 = ln z. Thus it remains to verify that (40) and (41) both have (bounded) classical solutions. But this can be done in a similar way as in the proof of the previous theorem with some modifications. We again refer to Ref. 9 for details. Finally, the explicit solution for the price p (39) follows from the principle of equivalent utility and solving the equation V 0 (t, w) = U (t, w + p, s, z). The proof is now complete.

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5. Some Remarks on General Insurance Models The results of previous sections can be extended to a more general life insurance model (see, e.g., Ref. 13). For example, we can consider the cases of “multiple decrement” with status-dependent payments, as well as the benefit be payable at a random time (such as “moment of death”), and so on. In these cases the the “state process” {Xt }t≥0 should modified as follows. Suppose that the policy starts at time 0 for a person aged x, and assume that X is a Markov chain with a finite state space {0, 1, ..., m}, representing the numerical code of the status at time t. We specify i = 1 to be a (absorbing) “cemetary state”, representing “death”, and X0 = 0 be the initial state. Now denote Iti = 1{Xt =i} , and define the counting process: Ntij = #{transitions of X from state i to j during [0, t]}. 4

(42)

Also, for each t we define a stopping time τt = inf{s ≥ t : Xs 6= Xt }; and τti = [1{Xτt =i} ]−1 τt , for i = 0, ..., m. Using these stopping times we can define the conditional probabilities: for i, j ∈ {0, ..., m}, 4 ¯is = tp

P {τs > t|Xs = i};

4 ¯sij = tq

P {τsj = τs ≤ t|Xs = i},

s ≤ t.

(43)

Similar to (1), we can define the “force of decrement of status i due to cause j” as 4 ¯ ij = λ lim t

¯tij t+h q

h→0

i, j = 0, 1, · · · m.

,

h

(44)

¯ ij is nothing but the transition It is not hard to show (see 9 for details ) that λ t intensity of the Markov chain X, defined as λij t = lim 4

h↓0

ij t+h qt

,

h

i 6= j,

(45)

4

where t qsij = P {Xt = j|Xs = i} is the transition probability. We note that if m = 1, then there are only two states: life or death. In this case the state process X becomes the one in the previous section, and the general life model reduces to the simple life model with τ01 = T (x). With a slight abuse of notation, we shall use 4

˜

the same notation F for the filtration F = {Ft }t≥0 = {FtX ∨ FtB ∨ FtB }t≥0 . Assume now in the general life insurance problem there are two types of payments: one takes the form of “annuity” and the other of “insurance”. More precisely, the (cumulative) payment process is XZ t XZ t At = Iui bi (u, Su , Zu )du + bij (u, Su , Zu )dNuij , t ≥ 0, (46) i

0

i6=j

0

where bi (t, s, z) is the rate of net payment of life annuity at state i, given St = s, Zt = z; and bij (t, s, z) is the rate of net payment of life insurance upon transition from state i to state j, given St = s, Zt = z. Given a payment process A, the wealth process (5) should then be modified to the following: dWtπ = rWtπ dt + πtT (µt − rt 1)dt + πtT σt dBt − dAt ,

t ≥ 0.

(47)

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Since the non-tradability of the asset Z will not play any essential role in such a setting, one can simply consider it as the (d + 1)-th component of the vector S = (S 1 , · · · , S d , S d+1 ). and write the SDE for the risky assets as dSt = D[St ]{µt dt + σt dBt },

t ≥ 0,

(48)

where D[s] denotes the diagonal matrix diag[s1 , ..., sd+1 ], and B is a d+1-dimensional Brownian motion. For a given portfolio π and the initial data (t, w, s), we denote W t,w,π (resp. S t,s ) to be the solution to (25) (resp. (48)), such that Wt = w (resp. St = s). We now introduce a family of value functions in accordance with all the status. For all (t, w, s) ∈ [0, T ] × IR × IRd+1 + , and k = 0, · · · , m, U k (t, w, s) = sup E{u(WTπ )|Wt = w, St = s, Xt = k}. 4

(49)

π∈Au

Then, by a standard procedure of dynamic programming, one can more or less conjecture that the HJB equation for the family {U k ; k = 1, · · · , m} should be a system of partial differential equations, but the exact form of these PDEs does not seem to be clear without a careful examination. To obtain the HJB equation, let us denote, for ϕ ∈ C 1,2 ([0, T ] × IRd+2 ), X X L[ϕ] = Lt,w,s [ϕ] = µit si ϕsi (t, w, s) + σtik σtjk si sj ϕsi sj (t, w, s) (50) i

i,j,k

2 = hϕs (t, w, s), D[s]µt i + tr{D[s](σt σtT )D[s](Dss ϕ(t, w, s))}.

Further, for (t, w, s) ∈ [0, T ] × IR × IRd+1 , (ϕ, ψ, p) ∈ IR × (−∞, 0) × IRd+1 , and k = 0, 1, · · · , m, we define  4 1   Hk (t, w, s, ϕ, ψ, p; π) = |σt π|2 ψ + [hπ, µt − rt 1i + rt w − bk (t, s)]ϕ   2  (51) +hπ, σt σtT D[s]pi     k 4  H (t, w, s, ϕ, ψ, p) = supπ Hk (t, w, s, ϕ, ψ, p; π).

Note out that the quadratic nature (in π) of the Hamiltonian in (51) and the unrestricted choice of π implies that H k < ∞ if and only if ψ ∈ (−∞, 0). We have the following theorem, whose proof can be found in Ref. 9. Theorem 5.1. Under some standard conditions on the coefficients, and assume that for all k = 0, 1, ..., m the value functions U k ∈ C 1,2,2 ([0, T ] × IR × IRn ). Then, for each k, U k is strictly concave in w, and U = (U 0 , U 1 , · · · , U m ) satisfies the following system of HJB equations: X kj  k k  0 = Utk + L[U k ]+H k (t, w, s, Uwk , Uww , Uws )+ λt (U j (t, w − bkj , s) − U k ), (52) j6=k  k U (w, T, s) = u(w), k = 0, 1, · · · , m.

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Finally, we remark here that although Theorem 5.1 gives only a sufficient condition, it shows a significant difference between this HJB equation and the traditional ones. A rigorous proof of the theorem is by no means trivial, due to the generality of the utility function (see Ref. 9). A more detailed study of the value function and the fact that the value function is indeed a viscosity solution to the HJB equation will be studied in another forthcoming paper 10 . References 1. Bielecki, T.R., Jeanblanc, M., and Rutkowski, M., Hedging of defaultable claims, Paris-Princeton Lectures on Mathematical Finance (2004). 2. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Actuarial Mathematics, The Society of Actuaries, USA (1997). 3. Frittelli, M., Introduction to a theory of value coherent with the no-arbitrage principle, Finance and Stochastics 4(3), 275-297 (2000). 4. Gerber, H.U., An Introduction to Mathematical Risk Theory, Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Phiadelphia (1979). 5. Hodges, S.D. and Neuberger, A., Optimal Replication of Contingent Claims under Transaction Costs, Rev. Futures Markets 8, 222-239 (1989). 6. Karatzas, I. and Shreve, S.E., Brownian Motion and Stochastic Analysis, SpringerVerlag (1988). 7. Ladyzenskaja, O.A., Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society (1968). 8. Ma, J. and Yong, J. Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, 1702, Springer (1999). 9. Ma, J. and Yu, Y. Principle of Equivalent Utilility and Universal Variable Life Insurance Pricing (2005), preprint. 10. Ma, J. and Yu, Y. UVL Insurance Pricing Problems and Systems of Partial Differential-Difference Equations, working paper. 11. Musiela, M. and Zariphopoulou, T. An Example of Indifference Prices under Exponential Preferences, Finance Stoch. 8, no. 2, 229-239 (2004). 12. Musiela, M. and Zariphopoulou, T. Indifference Prices of Early Exercise Claims, Mathematics of finance, 259–271, Contemp. Math., 351 (2004) Amer. Math. Soc., Providence, RI. 13. Norberg, R., Hattendorff’s Theorem and Thiele’s Differential Equation Generalized, Scand. Actuarial J. 1, 2-14 (1992). 14. Owen, M.P., Utility based optimal hedging in incomplete markets, Annals of Applied Probability 12, 691-709 (2002). 15. Protter, P., Stochastic integration and differential equations: A new approach, Springer-Verlag, Berlin (1990). 16. Rouge, R. and El Karoui, N., Pricing via utility maximization and entropy, Mathematical Finance 10, 259-276 (2000). 17. Yong, J and Zhou, X.Y., Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York (1999). 18. Young, V.R. and Zariphopoulou, T., Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility, Scand. Actuarial J. 4, 246-279 (2002).

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Chap20-PengShige

GΓ -EXPECTATIONS AND THE RELATED NONLINEAR DOOB-MEYER DECOMPOSITION THEOREM

SHIGE PENG∗ and MINGYU XU School of Mathematics and System Science, Shandong University, 250100, Jinan, China [email protected] (S. Peng), [email protected] (M. Xu)

In this paper, we define a nonlinear expectation via the BSDE with constraint (BSDE with sigular coefficient), then we introduce definitions of super and sub martingale under this nonlinear expectation and prove their nonlinear Doob-Meyer type decomposition

1. Introduction The objective of backward stochastic differential equation (BSDE in short) with generator g and with constraint Γ is to find the smallest (g–) supersolution (Y, Z, A), where A is an increasing process by which (Y, Z) is forced to remain inside of a given subset Γ ⊂ R × Rd. This problem was studied by El Karoui, Kapoudjian, Pardoux, Peng and Quenez, 16 for the constraint only on Y called reflected BSDE, and then initialed an extensive study (see Refs. 9, 19, 22, 24, 28, 29, 41, 42 among many others) for single and double barriers of reflected BSDEs. El Karoui & Quenez 17 and then Cvitanic & Karatzas, 7,8 Karatzas & Kou 25 may be considered as a strong motivation in finance to BSDE with constraint on Z. Cvitanic, Karatzas and Soner 10 studied this problem of BSDE for the case where the constraint Γ of Z is convex and the corresponding generator g is concave. Peng 38 studied constraints on (Y, Z) where g is a general Lipschitz function. As we will show in this paper, this result can applied to a very general situation where the corresponding Γ was assumed to be only a closed subset of R × Rd . In this paper we will work with these weaker conditions. This will provide a wide space of freedom to treat different types of situations. Typically, in the situation of differential games, the generator g is neither convex nor concave (see Refs. 20, 21 and 23). Recently, special constraint BSDEs (reflected BSDEs) are extended to the high dimensional case (see Refs. 32 and 35). ∗ The

author would like to acknowledge the partial support from the natural science foundation of china, grant no. 10131040. 122

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An interesting point of view is that this smallest Γ–constrained g-supersolution is, in fact, the solution of the BSDE with a singular coefficient gΓ defined by gΓ (t, y, z) = g(t, y, z)1Γ(y, z) + (+∞) · 1ΓC (y, z). Recent developments of continuous time finance requires a nonlinear version of conditional expectations. In 1997, the first author has introduced a Brownian filtration (Ft )t≥0 consistent nonlinear expectation Eg [X] : X ∈ L2 (Ω, FT , P ) → R call g–expectation, which is defined by y0X , where (ytX , ztX )0≤t≤T is the solution of the BSDE with a given Lipschitz function g(t, y, z), as its coefficient and with the above X as its terminal condition. Here we assume g satisfies Lipschitz condition in (y, z) as well as g(t, y, 0) ≡ 0. When g is a linear function in (y, z), this g–expectation Eg [·] is just a Girsanov transformation. But it becomes a nonlinear functional once g is nonlinear in (y, z), i.e., Eg [·] is a constant preserving monotonic and nonlinear functional defined on L2 (Ω, FT , P ). A significant feature of this nonlinear expectation is that, just like the classical notion of the conditional expectation, the g–expectation of X under Ft , noted by Eg [X|Ft ] ∈ L2 (Ω, Ft , P ), can be still characterized by Eg [X1A ] ≡ Eg [Eg [X|Ft ]1A ], ∀A ∈ Ft . In fact, it is proved that the unique element in L2 (Ω, Ft , P ) satisfying the above classical criterion is Eg [X|Ft ] = ytX . This stricken fact gives us a hint: many beautiful and powerful properties in the modern probability might still hold true without the linearity assumption. For example the essentially central notions of martingales, sub and supermartingales do not need the linearity. We then can ask if the well–known submartingale decomposition theorem is still hold true. In fact we have to introduce an intrinsic proof for those nonlinear decomposition theorem. (see Refs. 6, 38, 39, 40, and 44). Similarly as the above g–expectation, we can also define the corresponding g Γ – expectations by the smallest solution of BSDE with gΓ as well as the corresponding gΓ –supermartingales and submartingales. We then prove a gΓ –supermartingale decomposition theorem, which is a nonlinear version of Doob–Meyer decomposition theorem. The gΓ –submartingale decomposition can not be obtained by the above mentioned gΓ –supermartingale decomposition theorem. We shall obtain this decomposition theory in a quite different way. Recently a profound link between super–replication, risk measures (see Refs. 1 and 18) nonlinear expectations are being explored (see Refs. 3, 39, and 45). We hope that the results of this paper will be proved to be useful in this direction. We also refer to Refs. 2, 4, 5, 13, 14, 24, 30, and 31 for interesting research works in this domain. This paper is organized as follows. In the next section we list our main notations and main conditions required. In Section 3 we shall introduce the gΓ –expectations

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and related definitions. In Section 4, we will give the results of the existence and uniqueness of the solution of reflected BSDE with constraints. In Section 5, we present and prove the the nonlinear decomposition theorem of the related gΓ – supermartingales and gΓ –submartingales. Some useful results such as monotonic limit theorem, are listed in Appendix. 2. gΓ -Solution: the Smallest Supersolution of BSDE Constrained in Γ Let (Ω, F, P ) be a probability space, and B = (B1 , B2 , · · · , Bd )T be a d-dimensional Brownian motion defined on [0, ∞). We denote by {Ft ; 0 ≤ t < ∞} the natural filtration generated by this Brownian motion B : Ft = σ{{Bs ; 0 ≤ s ≤ t} ∪ N }, where N is the collection of all P −null sets of F. The Euclidean norm of an element x ∈ Rm is denoted by |x|. We also need the following notations for p ∈ [1, ∞): • Lp (Ft ; Rm ) :={Rm -valued Ft –measurable random variables X s.t. E[|X|p ] < ∞}; • LpF (0, t;RRm ) :={Rm –valued and Ft –adapted processes ϕ defined on [0, t], t s. t. E 0 |ϕs |p ds < ∞}; • DpF (0, t; Rm ) :={Rm –valued and RCLL Ft –progressively measurable processes ϕ on [0, t], s.t. E[sup0≤s≤t |ϕs |p ] < ∞}; • SpF (0, t; Rm ) :={continuous processes in DpF (0, t; Rm )}; • ApF (0, t) :={increasing processes in DpF (0, t; R) with A(0) = 0}. When m = 1, they are denoted by Lp (Ft ), LpF (0, t), DpF (0, t) and SpF (0, t), respectively. We are mainly interested in the case p = 2. In this section, we consider BSDE on the interval [0, T ], with a fixed T > 0. We are given a function g(ω, t, y, z) : Ω × [0, T ] × R × Rd → R, which always plays the role of the coefficient of our BSDEs. g satisfies the following assumption: there exists a constant µ > 0, such that, for each y, y 0 in R and z, z 0 in Rd , we have (i) g(·, y, z) ∈ L2F (0, T ), (ii) |g(t, ω, y, z) − g(t, ω, y 0 , z 0 )| ≤ µ(|y − y 0 | + |z − z 0 |).

(1)

The constraint of our BSDE is Γ(t, ω) : Ω×[0, T ] → C(R × Rd ), where C(R × Rd ) is the collection of all closed non–empty subsets of R × Rd which is Ft –adapted, namely, (i) (y, z) ∈ Γt (ω) iff dΓt (ω) (y, z) = 0, t ∈ [0, T ], a.e.; (ii) dΓ· (y, z) is Ft –adapted process, for each (y, z) ∈ R × Rd ,

(2)

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where dΓ (·) is a distant function of Γ: dΓt (y, z) :=

inf

(y 0 ,z 0 )∈Γt

(|y − y 0 |2 + |z − z 0 |2 )1/2 ∧ 1.

dΓt (y, z) is a Lipschitz function: for each y, y 0 in R and z, z 0 in Rd , we always have |dΓt (y, z) − dΓt (y 0 , z 0 )| ≤ (|y − y 0 |2 + |z − z 0 |2 )1/2 . Remark 2.1. The constraint discussed in Peng38 is Γt (ω) = {(y, z) ∈ R1+d : Φ(ω, t, y, z) = 0}.

(3)

Here Φ(ω, t, y, z) : Ω × [0, T ] × R × Rd → [0, ∞) is a given nonnegative measurable function, and satisfies integrability condition and Lipschitz condition. In this paper we always consider the case Φ(t, y, z) = dΓt (y, z). We are then within the framework of supersolution of BSDE of the following type: Definition 2.1. (g–supersolution, cf. El Karoui, Peng and Quenez15 (1997) and Peng38 (1999) ) A process y ∈ D2F (0, T ) is called a g–supersolution if there exist a predictable process z ∈ L2F (0, T ; Rd) and an increasing RCLL process A ∈ A2F (0, T ) such that Z T Z T yt = y T + g(s, ys , zs )ds + AT − At − zs dBs , t ∈ [0, T ]. (4) t

t

Here z and A are called martingale part and increasing part, respectively. y is called a g–solution if At ≡ 0. y is called a Γ–constrained g–supersolution if y and its corresponding martingale part z satisfy (yt , zt ) ∈ Γt , (or dΓt (yt , zt ) = 0), dP × dt a.s. in Ω × [0, T ].

(5)

Remark 2.2. We observe that, if y ∈ D2F (0, T ) is a g–supersolution, then the pair (z, A) in (4) are uniquely determined since the martingale part z is uniquely determined. Occasionally, we also call (y, z, A) a g–supersolution. By Peng38 , (see Appendix Theorem 6.2), if there exists at least one Γ– constrained g–supersolution, then the smallest Γ–constrained g–supersolution exists. In fact, a Γ-constraint g-supersolution can be considered as a solution of the BSDE with a singular coefficient gΓ defined by gΓ (t, y, z) = g(t, y, z)1Γt (y, z) + (+∞) · 1ΓCt (y, z). So we define the smallest Γ–constrained g–supersolution by gΓ –solution.

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Definition 2.2. (gΓ –solution) y is called the gΓ –solution on [0, T ] with a given terminal condition X if it is the smallest Γ–constrained g–supersolution with yT = X: for t ∈ [0, T ], Z T Z T yt = X + g(s, ys , zs )ds + AT − At − zs dBs , (6) t

t

(yt , zt ) ∈ Γt , dAt ≥ 0. Remark 2.3. (y, z, A) is the gΓ -solution does not implies that the increasing pro¯ with cess is also the smallest one, i.e. for any Γ–constrained g–supersolution (¯ y , z¯, A) ¯ the same terminal X, we may have At ≥ At . An example is as follows: consider the case [0, T ] = [0, 2], X = 0, g = 0 and Γt = {(y, z) : y ≥ 1[0,1] (t)}. It’s easy to see that the smallest solution is yt = 1[0,1) (t) with zt = 0, At = 1[1,2] (t). Obviously y t = 1[0,2) (t) with z t = 0, At = 1{t=2} (t) is another Γ–constrained g–supersolution with the same terminal condition yT0 = 0. But we have At > At on the interval [1, 2). 3. F-Consistent Nonlinear Expectations 3.1. gΓ –expectations We now introduce a notion of F–consistent nonlinear expectations via gΓ –solutions. We make the following assumption: there exists a large enough constant C0 such that g(t, y, 0) ≤ C0 + µ|y|, ∀y ≥ C0 , and (y, 0) ∈ Γt , ∀y ≥ C0

(7)

We need the terminal conditions to be in the following linear subspace of L2 (FT ): L2+,∞ (FT ) := {ξ ∈ L2 (FT ), ξ + ∈ L∞ (FT ). Proposition 3.1. We assume (1) and (7). Then for each X ∈ L2+,∞ (FT ), the smallest Γ–constrained g–supersolution with terminal condition y T = X exists. Furthermore, we have yt ∈ L2+,∞ (FT ), t ∈ [0, T ]. Proof. We consider

y0 (t) = ( X + ∞ ∨ C0 )eµ(T −t) + C0 (T − t) + (X − X + ∞ ∨ C0 )1{t=T } .

It is the solution of the following backward equation: Z T y0 (t) = X + (C0 + µ|y0 (s)|)ds + A0 (T ) − A0 (t), t

0

where A is an increasing process: A0 (t) := (kX + k∞ ∨ C0 − X)1t=T . But y0 (·) can be expressed to: Z T Z T y0 (t) = X + g(s, y0 (s), 0)ds + [c + µ|y0 (s)| − g(s, y0 (s), 0)]ds + A0 (T ) − A0 (t). t

t

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Thus the triple defined on [0, T ] by (y1 (t), z1 (t), A1 (t)) := (y0 (t), 0,

Z

t

[c + µ|y0 (s)| − g(s, y0 (s), 0)]ds + A0 (t)) 0

is a Γ–constrained g–supersolution with y1 (T ) = X. According to Theorem 6.2, the gΓ –solution with y(T ) = X exists. We also have (yt )+ ∈ L∞ (FT ) since yt ≤ y1 (t) = y0 (t). We now introduce the notion of gΓ –expectation Definition 3.1. We assume that g(t, 0, 0) = 0 and (0, 0) ∈ Γt , for each 0 ≤ t ≤ T < ∞. Then consider X ∈ L2 (FT ) with X + ∈ L∞ (FT ), let (y, z, A) be the gΓ – Γ solution defined on [0, T ] with terminal condition yT = X. We set Egt,T [X] := yt . The system Γ [·] : L2+,∞ (FT ) → L2+,∞ (Ft ), Egt,T

0≤t≤T 0,

and consequently P (t) ≤ P (0) e−0 t + C1 /0 ,

t ≥ 0.

(10)

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On the other hand, from (6) with = 1 and ρ = ρ0 , we get Z 1 1 2 2 2 |ux |4 dx P (t) ≥ ρ0 kut k + ρ0 kuxx k − ρ0 β kux k + ρ0 κ 2 42 Ω 1 1 7γ 2 ρ0 + δ− − ρ0 kuk2 − ρ0 kf k2 − 2 2ρ0 κ 1 2 2 ≥ ρ0 kut k + ρ0 kuxx k − C2 , 2

(11)

where C2 =

21 2ρ0 κ

ρ0 |β| +

2 p 1 1 7γ 2 ρ0 λ1 δ − − ρ0 + ρ0 kf k2 . − 2 2ρ0 κ

Substituting (11) into (10), we find that kw(t)k2E ≤

2 2 (P (t) + C2 ) ≤ P (0) e−0 t + C1 /0 + C2 , ρ0 ρ0

t ≥ 0,

(12)

which implies that lim sup kw(t)k2E ≤ 2 (C1 + C2 0 ) .

(13)

t→∞

Finally, (12) implies that the solution w(t) = w (t; w0 ) exists globally on [0, ∞) and (13) in further implies that any bounded ball Br0 with r0 > 2 (C1 + C2 0 ) is an absorbing ball for this solution semigroup S(t), t ≥ 0. 2. The Existence of a Global Attractor We take the precompact pseudometric approach to prove the asymptotical compactness of the solution semigroup S(t) of the abstract Boussinesq equation (2), which together with the shown absorbing property implies the existence of a global attractor for S(t). Let us introduce a functional 2

Π(u, v) = kuxx k +

1 δθkuk2 + θhu, vi + kvk2 , 2

where θ > 0 is a small parameter. For ϕ = (u, v)T ∈ E, Π(u, v) is equivalent to T kϕk2E . For any two solutions wi = (ui (t), ∂t ui ) of the equation (2), i = 1, 2, we shall denote the difference by W (t) = w1 (t) − w2 (t) and accordingly U (t) = u1 (t) − u2 (t). Lemma 2.1. For any given bounded set M ⊂ E, there exists a constant θ > 0 and a constant L0 = L0 (M, θ) > 0 such that for any two solutions as above of the equation (2), with wi (0) ∈ M , i = 1, 2, it holds that Π (U (t), Ut (t)) ≤ e−θt Π (U (0), Ut (0)) + L0 sup kU (s)k + sup kUx (s)kC[0,1] 0≤s≤t

0≤s≤t

(14) for t ≥ 0.

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Proof. Since Wi (t), i = 1, 2, are strong solutions of (2), the function U (t) satisfies the equation 2

Utt + δUt + Uxxxx = −βUxx + κ (u1x ) Uxx + κ (u1x + u2x ) Ux u2xx −2γ (u1xx U + u2 Uxx ) − 2γ (u1x + u2x ) Ux .

(15)

Taking the inner product in H of (15) with 2Ut + θU and using the imbedding j H j+1 (Ω) ⊂ CB (Ω) for j = 0, 1, we can conduct an a priori estimate to reach the inequality 1 d kUt k2 + kUxx k2 − β kUx k2 + θ hUt , U i + θδkU k2 dt 2 D E 2 2 2 2 2 (16) +κ ku1x Ux k − 2γ u2 , (Ux ) + (2δ − θ) kUt k + θ kUxx k − θβ kUx k ≤ L1 kU (t)k + L2 kUx (t)kC[0,1] ,

where 0 < θ ≤ 1 and L1 = L1 (M ), L2 = L2 (M ) are two positive constants only depending on M . Define another functional D E 2 2 2 Θ(u, v) = Π(u, v) − β kux k + κ ku1x ux k − 2γ u2 , (ux ) , where u = u1 − u2 . Then we have d Θ (U (t), Ut (t)) + θΘ (U (t), Ut (t)) dt 1 2 2 2 2 ≤ θ kUt k + θ kUxx k + θ2 hU, Ut i + θ2 δkU k2 + (θ − 2δ) kUt k − θ kUxx k 2 D E +θκ ku1x Ux k2 − 2θγ u2 , (Ux )2 + L1 kU k + L2 kUx kC[0,1] .

Choose 0 < θ ≤ min{1, δ} and note that M is bounded in E. We get d Θ (U (t), Ut (t)) + θΘ (U (t), Ut (t)) ≤ L3 kU (t)k + L4 kUx (t)kC[0,1] , dt

(17)

where L3 and L4 are uniform constants. Integrating (17) and back to Π (U (t), Ut (t)), finally we can reach the inequality (14), with 0 < θ ≤ min{1, δ/2}. Theorem 2.1. There exists a global attractor A for the solution semigroup S(t) generated by the abstract Boussinesq equation (2) on E. Proof. By Lemma 2.1, with θ = δ/2 in (14), we see that Π(u, v) is equivalent to

2 the norm square (u, v)T E in the following sense,

1

(u, v)T 2 ≤ Π(u, v) ≤ max 1 + 3 δ 2 λ−1 , 3 (u, v)T 2 , (18) 1 E E 2 8 2

in which λ1 is the smallest eigenvalue of the operator A. Then (14) and (18) imply that there exist two constants C > 0 and L > 0 only depending on the original

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parameters in (1), such that kS(t)w10 − S(t)w20 k ≤ Ce−δt/4 kw10 − w20 kE 1/2 1/2 +L sup kU (s)k + sup kUx (s)kC[0,1] . 0≤s≤t

(19)

0≤s≤t

For each given t > 0, define ρt (w10 , w20 ) = ρ1t (w10 , w20 ) + ρ2t (w10 , w20 ) where ρ1t (w10 , w20 ) = L sup kU (s)k1/2 = L sup ku1 (s) − u2 (s)k 0≤s≤t

1/2

0≤s≤t

and 1/2

1/2

ρ2t (w10 , w20 ) = L sup kUx (s)kC[0,1] = L sup k∂x u1 (s) − ∂x u2 (s)kC[0,1] . 0≤s≤t

0≤s≤t

Now we show that ρt is a precompact pseudometric on E = V × H. First we show that ρ1t is a precompact pseudometric on E for each given t > 0. T Suppose that {w0n } is a bounded sequence in E and let wn (t) = (un (t), ∂t un (t)) be n the solution of (2) with w0 . From the boundedness of solutions shown in Theorem 1.1, there is a constant K > 0 such that sup kun (t)kH 2 (0,1) + sup k∂t un (t)kL2 (0,1) ≤ K. t≥0

0

(20)

t≥0

Hence, {un (s)} for each s ∈ [0, t] is in a compact set of H = L2 (0, 1). On the other hand, since {∂t un (t)} is uniformly bounded, by the imbedding V = H02 (0, 1) ⊂ C 1 [0, 1], and by the Leibniz formula, for a given t > 0, there is a constant K1 > 0 such that kun (t1 ) − un (t2 )k ≤ K1 |t1 − t2 | , for any t1 , t2 ∈ [0, t]. Therefore by the Arzel´ a-Ascoli theorem in Banach space, we can assert that {un (·)} is a precompact set in C [0, t]; L2 (0, 1) ,

so that there exists a convergent (also Cauchy) subsequence with respect to the pseudometric ρ1t . By definition, cf. Sell and You, 10 ρ1t is precompact on E. Next we prove that ρ2t is also a precompact pseudometric. Let {wn } be as above. We want to show that {∂x un (·)} is a precompact set in C([0, t]; C[0, 1]).

(21)

Note that C([0, t]; C[0, 1]) = C([0, t] × [0, 1]) = C([0, 1]; [0, t]) and their norms are equivalent. It reduces to proving {∂x un (·)} is a precompact sequence in C([0, 1]; [0, t]).

(22)

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By (20) and the Leibniz formula, there is a constant K2 > 0 such that k∂x un (σ1 ) − ∂x un (σ2 )kC[0,t] ≤ K2 |σ1 − σ2 | , for any σ1 , σ2 (spatial variable) in [0, 1]. Moreover, for each given 0 ≤ σ ≤ 1, we claim that {∂x un (·)} is precompact in C[0, t].

(23)

In order to verify (23), it suffices to check the following two properties. (i) For each given 0 ≤ σ ≤ 1, there exists a constant b1 (σ, t) such that |∂x un (σ, s)| ≤ b1 (σ, t), for all s ∈ [0, t] and for n ≥ 1. This property is satisfied because |∂x un (σ, s)| ≤ k∂x un (σ, ·)kC[0,t] ≤ const sup kun (·, s)kV ≤ C(t), 0≤s≤t

for all s ∈ [0, t], n ≥ 1, based on the boundedness of {w0n } and the absorbing property. Then we can take b1 (σ, t) = C(t), which is even independent of σ ∈ [0, 1]. (ii) For each given 0 ≤ σ ≤ 1, there exists a constant b2 (σ, t) such that |∂x un (σ, s1 ) − ∂x un (σ, s2 )| ≤ b2 (σ, t) |s1 − s2 | , for all s1 , s2 ∈ [0, 1] and for n ≥ 1. Below we prove this inequality. Indeed, Z t wn (t) = e−Gt w0n + e−G(t−s) F (wn (s)) ds,

t ≥ 0,

(24)

0

in which e−Gt is a differential semigroup as said earlier. It follows that e−Gt w0n is a classical solution of the homogeneous equation and Z t −Gt e w0 − w 0 = −Ge−Gs w0 ds, t ≥ 0, w0 ∈ E. (25) 0

{w0n } is a bounded sequence in E, from (25) it of e−Gt w0n is uniformly Lipschitz continuous

Since yn (t) Thus there exists a constant d1 (t) such that

is seen that the first component on any compact time interval.

kyn (s1 ) − yn (s2 )kV ≤ d1 (t) |s1 − s2 | , for any s1 , s2 ∈ [0, t],

n ≥ 1.

(26)

Substituting (26) into (24) and noting that F ∈ CLip (E), we can deduce that there exists a constant d2 (t) > 0 such that kun (·, s1 ) − un (·, s2 )kV ≤ d2 (t) |s1 − s2 | , for any s1 , s2 ∈ [0, t],

n ≥ 1.

In this step, Gronwall inequality is used. Then it follows that the inequality claimed in this item (ii) is valid with b2 (σ, t) = d2 (t), again independent of σ ∈ [0, 1]. Finally we can synthesize what has been proved: The items (i) and (ii) ensure that (23) holds, which together with the equicontinuity implies that (22) and (21) are valid. Hence, ρ2t is a precompact pseudometric on E. It follows that ρt is a precompact pseudometric on E. By Lemma 22.5 in Sell and You10 , (19) then

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implies that the solution semiflow {S(t)}t≥0 of the abstract Boussinesq equation (2) is κ-contracting. According to Theorem 23.12 in Sell and You 10 on the existence of a global attractor, we can conclude that a global attractor A exists in E for this solution semiflow. 3. Hausdorff Dimension of the Global Attractor Since the global attractor A ⊂ E is a nonempty, compact, invariant set, one has S(t) A = A for any t ≥ 0. A straightforward but tedious processing can verify that the nonlinear mapping S(t) of the solution semigroup of the equation (2) is continuously Fréchet differentiable for each t ≥ 0. Here we use the technique of isomorphic re-installation of solutions. Let ϕ = (u, v)T , v = ut + u, where is an arbitrary positive constant. Then (2) can be rewritten as dϕ + Λϕ = F (ϕ), t > 0, dt (27) T ϕ(0) = ϕ0 = (u0 , u1 + u0 ) , where Λ=

I A + ( − δ) I

−I (δ − ) I

: D(A) × V → E

and F : E → E remains the same as in (2). For constants σ1 , σ2 > 0 and ϕ ∈ D(Λ) = D(A) × V , we can calculate hΛϕ, ϕiE − σ1 kϕk2E − σ2 kvk2 2

−1/2

≥ ( − σ1 ) kuxx k − λ1

|( − δ)| kuxx k kvk + (δ − − σ1 − σ2 ) kvk2 .

If we choose , σ1 , σ2 such that δ ≥ 2,

and σ1 = σ2 =

1 , 4

then for sufficiently small > 0, one can get hΛϕ, ϕiE ≥ σ1 kϕk2E + σ2 kvk2 .

(28)

Now the solution mapping SE (t) : E → E defined by the equation (27) is relative to the solution mapping S(t) defined by (2) by the following reversible transformation S (t) = R S(t) R− ,

(29)

where R is the isomorphism of E given by R : (u, v)T → (u, v+u)T . Consequently R A is the global attractor of the semiflow S (T ), t ≥ 0. Moreover, we can check that A and R A have the same Hausdorff dimension. The linear variational equation of (27) is dΨ = (−Λ + DF (ϕ)) Ψ, dt Ψ(0) = (ξ, η)T ∈ E,

(30)

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where Ψ = (U, V )T . In (30), ϕ = (u, v)T is a particular solution of (27), and 0 DF (ϕ) Ψ = . −βUxx + 2κuxxux Ux + κ (ux )2 Uxx − 2γ(uU )xx From (29) it is clear that for t > 0, S (t) is Fréchet differentiable on E. Its T Fréchet derivative at ϕ0 = (u0 , u1 + u0 ) is the linear solution operator: (ξ, η)T → (U (t), V (t))T of the variation equation (30). Lemma 3.1. Let Φ = {Φ1 , Φ2 , . . . , Φ` } be a set of ` orthonormal vectors in E. If the following condition is satisfied, sup sup

` X

Φ⊂E ϕ∈R A j=1

h(−Λ + DF (ϕ)) Φj ; Φj iE ≤ 0,

(31)

then the Hausdorff dimension of the global attractor R A for the semiflow S (t), t ≥ 0, is less than or equal to `. Proof. This is a direct consequence of Theorem V.3.3 in Temam

12

.

Theorem 3.1. The Hausdorff dimension dH (R A) of the global attractor R A for the semiflow S (t) generated by (27) on E satisfies   `  1X M02  dH (R A) ≤ min ` ∈ N : , (32) λj ≥  ` j=1 4σ1 σ2  for some constant M0 , where λj , j = 1, . . . , `, are the first ` eigenvalues of the operator A, and constants σ1 , σ2 satisfy (28).

Proof. Let ` ∈ N be fixed. Consider ` solutions Ψ1 , Ψ2 , . . . , Ψ` of the IVP (30). At a given time τ , let Q` (τ ) be the orthogonal projection of E onto the subspace T spanned by Ψ1 (τ ), Ψ2 (τ ), . . . , Ψ` (τ ). Let Φj (τ ) = (ξj , ηj ) ∈ E, j = 1, . . . , `, be an orthonormal basis of Q` (τ ) E. For ϕ ∈ R A, study ` X

h(−Λ + DF (ϕ)) Φj , Φj iE = −

j=1

` X

hΛΦj , Φj iE +

j=1

` X

h(DF (ϕ)) Φj , Φj iE .

j=1

From (28), we have 2

2

− hΛΦj , Φj iE ≤ −σ1 kξjxx k − σ2 kηj k . Besides we have

(33)

2 hDF (ϕ) Φj , Φj iE ≤ −βξjxx + 2κuxuxx ξjx + κ (ux ) ξjxx − 2γ (uξj )xx kηj k = kDB(u) ξj k kηj k ,

where 2

B(u) = −βuxx + κ (ux ) uxx − γ u2

xx

+f

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and DB(u) is the Fréchet derivative of B at u. Since F ∈ CLip (E) ∩ CF1 (E), where the notation is explained in Sell and You

10

, so that

B ∈ CLip (V, H) ∩ CF1 (V, H), and R A is a bounded set, there exist a constant M0 > 0 such that for ϕ ∈ R A, kDB(u) ξj k ≤ M0 kξjxx k . Hence we get hDF (ϕ) Φj , Φj iE ≤

M02 2 2 kξjxx k + σ2 kηj k . 4σ2

(34)

Using (33) and (34), we obtain h(−Λ + DF (ϕ)) Φj , Φj iE ≤ 2

M02 2 − σ1 kξjxx k 4σ2

2

where kξjxx k ≤ kΦj kE = 1, and on the other hand ` X

2

kξjxx k =

j=1

` X

hAξj , ξj i ≥

j=1

` X

λj .

j=1

Therefore, ` X

h(−Λ + DF (ϕ)) Φj , Φj iE ≤ −σ1

j=1

` X

λj +

j=1

M02 `. 4σ2

Now if we choose ` ∈ N such that `

1X M02 λj ≥ , ` j=1 4σ1 σ2 we then get ` X

h(−Λ + DF (ϕ)) Φj , Φj iE ≤ 0.

(35)

j=1

By Lemma 3.1, (35) implies that (32) holds. Theorem 3.2. The Hausdorff dimension dH (A) of the global attractor A for the abstract Boussinesq equation (2) in E admits the same finite upper bound as given by (32). This conclusion follows from Theorem 3.1 and the fact dH (A) = dH (R A).

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4. The Relative Lattice Dynamics In the recent decade, the dynamics of lattice systems have been studied by many authors, cf. Chow et al. 4 and references therein. Lattice differential equations arise in a variety of applications such as in image processing, pattern recognition, material science, and molecular and cell biology. As another important aspect, lattice dynamical systems also emerge from spatial discretization of evolutionary PDEs. Here we shall study a lattice dynamical system relative to the Boussinesq equations. The objective remains to explore the existence of a global attractor for this lattice system and to address the upper semicontinuity with respect to finitedimensional approximate systems. Consider the following second-order nonlinear lattice differential equation, dui d 2 ui +δ + (ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 ) dt dt i 1 h 3 3 +β (ui+1 − 2ui + ui−1 ) + λui − κ (ui+1 − ui ) − (ui − ui−1 ) = fi , 3

i ∈ Z, (36)

with the initial conditions ui (0) = ui0 ,

u˙ i (0) = ui1 ,

i ∈ Z,

(37)

where δ, λ, and κ are positive constants, β is a real constant, f = (fi ) ∈ `2 . The equation (36) can be regarded as a spatial discretization of the damped Boussinesq equation with γ = 0 and on the unbounded domain Ω = (−∞, ∞). We can allow γ 6= 0 but the treatment will be more complicated. Here we assume that λ > 4 |β|.

(38)

To formulate the IVP of the lattice differential equation (36)–(37) into an abstract evolutionary equation and IVP in the Hilbert space `2 , we define the following linear operators from `2 into itself: for u = (ui ) i ∈ Z ∈ `2 , (DU )i = ui+1 − ui ,

(D∗ U ) = ui − ui−1 ,

(BU )i = ui+1 − 2ui + ui−1 , (AU )i = ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 ,

i ∈ Z.

We can check that all these are bounded linear operators on `2 , and we have A = B2,

B = DD∗ = D∗ D.

For u = (ui )i∈Z , v = (vi )i∈Z in `2 , both bilinear forms h·, ·i and h·, ·iλ defined by X hu, vi = ui vi , hu, viλ = hBu, Bvi + λhu, vi i∈Z

are inner-products of the space `2 . Moreover, their induced norms are equivalent λkuk2 ≤ kuk2λ ≤ (16 + λ) kuk2 .

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We shall denote the Hilbert space `2 with the inner-product h·, ·iλ and the norm k · kλ specifically by `2λ . Define E = `2λ × `2 , which is endowed with the inner-product and the norm as follows: for ϕj = u(j) , v (j) ∈ E, j = 1, 2, E E D D hϕ1 , ϕ2 iE = u(1) , u(2) + v (1) , v (2) λ X (1) (2) (1) (2) (2) (1) + λui ui + vi vi , Bu = Bu i

i

i∈Z

and kϕk2E = hϕ, ϕiE , for ϕ ∈ E. With the above defined, (36)–(37) can be written as the following equation in E, d2 u du 1 +δ + Au + βBu + λu − κD (D∗ u)3 = f, t > 0, 2 dt dt 3 u(0) = (ui0 )i∈Z = u0 , u(0) ˙ = (ui1 )i∈Z = u1 ,

(39)

where u = (ui )i∈Z and f = (fi )i∈Z . Let v = u+u, ˙ where > 0 is a small parameter. We can choose > 0 sufficiently small such that 2 3 δ + − δ ≤ 0, (1 + δ) + 4|β| − λ ≤ 0, and − 2 ≥ 0, 2 2 2

(40)

due to (38). The IVP (39) can be further formulated into the following IVP in the space E, ϕ˙ + Γϕ = R(ϕ),

ϕ(0) = ϕ0 = (u0 , v0 )T = (u0 , u1 + u0 )T ∈ E,

(41)

where ϕ = (u, v)T , v = u˙ + u, u − v Γ(ϕ) = , Au + λu + (δ − )(v − u) and R(ϕ) =

0 , g(u)

with 1 g(u) = −βBu + κD (D∗ u)3 + f. 3 Lemma 4.1. For any given initial data ϕ0 = (u0 , v0 )T ∈ E, there exists a unique local solution ϕ(t) = (u(t), v(t))T of the IVP (41), such that ϕ ∈ C([0, T ), E) for some T > 0.

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Proof. By using the inequality (a + b)2 ≤ 2 a2 + b2 , one can verify that kBuk ≤ 4kuk,

and

kAuk ≤ 16kuk,

kDuk ≤ 2kuk,

2 X

3 2 (ui+1 − 2ui + ui−1 )

D (D∗ u) = i∈Z

h

2

× (ui+1 − ui ) + (ui+1 − ui ) (ui − ui−1 ) + (ui − ui−1 )

2

i2

(42)

.

Since ui → 0 as i → ∞, there exists an integer N = N (u) such that 2

2

(ui+1 − ui ) + (ui+1 − ui ) (ui − ui−1 ) + (ui − ui−1 ) ≤ 1, for |i| ≥ N + 1. Then we have

2 2 X

3 3 D (D∗ u) + 16kuk2.

D (D∗ u) ≤ i

(43)

|i|≤N

Moreover, we can check that there exists a uniform constant L1 = L1 () > 0 such that kΓϕ1 − Γϕ2 kE ≤ L1 kϕ1 − ϕ2 kE

for any ϕ1 , ϕ2 ∈ E.

(44)

For the nonlinear mapping R, we have

2

kR (ϕ1 ) − R (ϕ2 )k2E = g u(1) − g u(2)

2 2 3 3

2

∗ (1) ∗ (2)

≤ 2β 2 B u(1) − u(2) + κ2 D D u − D u

9

3 3 2 2 8 X (1) (1) (2) (2) 2 2 (1) (2) ≤ 2β B u − u ui − ui−1 − ui − ui−1 ,

+ κ 9 i∈Z

where

3 3 2 X (1) (1) (2) (2) ui − ui−1 − ui − ui−1 i∈Z

2 2 2 2 2 X (1) (2) (1) (2) (1) (1) (2) (2) ≤6 ui − u i + ui−1 − ui−1 ui − ui−1 + ui − ui−1 . i∈Z

For any given bounded set S ⊂ E, there is a constant L2 = L2 (S) such that for any ϕ = (u, v)T ∈ S, one has kϕkE ≤ L2 . From the above two inequalities it follows that

2 2

kR (ϕ1 ) − R (ϕ2 )k2E ≤ L3 B u(1) − u(2) + λ u(1) − u(2) (45) 2 ≤ L3 kϕ1 − ϕ2 kE , where L3 = max 2β 2 , 342 κ2 λ−1 L42 .

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Thus both Γ and R are locally Lipschitz continuous. Then by the existence theory of local solutions of abstract differential equations in Banach spaces, the conclusion holds. Lemma 4.2. Assuming (38) and (40) are satisfied, then for any f ∈ `2 and any initial data ϕ0 ∈ E, the solution ϕ(t) of the IVP (41) exists globally for t ≥ 0. Furthermore, there exists an absorbing ball for this solution semiflow S (t), t ≥ 0. Proof. Note that D, B, and A satisfy the following properties: for u, v ∈ `2 , hDu, vi = − hu, D∗ vi , hBu, vi = −hDu, Dvi, hAu, vi = hBu, Bvi. Let ϕ(t) = (u(t), v(t))T be a solution of the equation (41), where v(t) = u(t)+u(t), ˙ and > 0 satisfies (40). Taking the inner-product of (39) with v(t) in `2 , we obtain d M (t) + Q(t) = 0, dt

t > 0,

(46)

where M (t) =

2 1 1 β λ 1

kuk ˙ 2 + kBuk2 − kDuk2 + kuk2 + κ (D∗ u)2 2 2 2 2 12 δ −hf, ui + hu, ˙ ui + kuk2 2

and Q(t) = (δ − ) kuk ˙ 2 + kBuk2 − β kDuk2 + λ kuk2

2

+ κ (D∗ u)2 − hf, ui. 3

With the choice of which satisfies (40), we have 3 β 2 M (t) − Q(t) = − δ kuk ˙ − kBuk2 + kDuk2 2 2 2

2 + (δ − λ) kuk2 − κ (D∗ u) + 2 hu, ˙ ui 4 2 2 3 2 + − δ kuk ˙ + ((1 + δ) + 4 |β| − λ) kuk2 ≤ 0. ≤ 2 2 2

(47)

From (46) and (47), it follows that M (t) ≤ M (0) e−t ,

t ≥ 0.

For any given α > 0, we have 1 1 2 M (t) ≥ kuk ˙ + kBuk2 + 4 2

λ δ α 2 2 + − 2 − − 2 |β| kuk2 − kf k2. 2 2 8 α

(48)

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Since (38) implies that there is a constant α > 0 such that α λ 1− − 2α − 4 |β| ≥ 0, 4λ one can deduce that 1 1 2 2 M (t) ≥ kuk ˙ + kBuk2 + λα kuk2 − kf k2 , 4 2 α where the last inequality of (40) is used. Hence, 2 kBuk2 + λ kuk2 + kuk ˙ ≤ max 4, α−1 M (t) + 2α−1 kf k2 .

(49)

(50)

Combining (48) and (50), we end up with

2 kϕ(t)k2E = kBuk2 + λ kuk2 + kuk ˙ ≤ max 4, α−1 M (0) e−t + 2α−1 kf k2 ,

and finally

lim sup kϕ(t)k2E ≤ (const)kf k2 .

(51)

t→∞

Therefore, for any given ϕ0 ∈ E, the solution ϕ(t) of the IVP (41) exists globally for t ≥ 0. The ball Or0 centered at the origin and of radius r0 > (const)kf k2 is an absorbing ball in E for this solution semiflow S (t), t ≥ 0. Next we adopt the approach of “tail ends” estimation to show the asymptotic compactness of this solution semigroup. This approach was introduced in Bates et al.1 and used in Zhou, 15 which seems effective in dealing with the semiflows on unbounded domains. Lemma 4.3. Under the same assumptions as in Lemma 4.2, let Or0 be the bounded absorbing ball given in the proof of Lemma 4.2. Then for any η > 0, there exist constants K(η) > 0 and T (η) > 0 such that for any initial data ϕ0 =(u0 , v0 )T = T

(u0 , u1 + u0 ) ∈ Or0 , the solution ϕ(t) = (ϕi (t))i∈Z = (ui (t), vi (t))

T

i∈Z

satisfies the following estimate, X X 2 2 2 2 |ϕi (t)|E , |(Bu(t))i | + λ |ui (t)| + |vi (t)| ≤ η, i≥K(η)

of (41) (52)

i≥K(η)

for all t ≥ T (η). Proof. Take a truncating function θ ∈ C 1 (R+ ) which satisfies θ(t) = 0,

for 0 ≤ t ≤ 1,

0 ≤ θ(t) ≤ 1,

for 1 ≤ t ≤ 2,

θ(t) = 1, for t ≥ 2. T For a solution ϕ(t) = (ϕi (t))i∈Z = (ui (t), vi (t)) u˙ i (t) + ui (t). Let m be a fixed integer. Set

w = (wi )i∈Z , where wi (t) = θ

|i| m

i∈Z

of (41), one has vi (t) =

ui (t),

i ∈ Z.

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By taking the inner product of the equation (41) with w˙ + w, we can get d Mθ (t) + Qθ (t) = 0, dt in which

t > 0,

(53)

X |i| 1 1 β λ 2 2 2 2 Mθ (t) = θ |u˙ i | + |(Bu)i | − |(Du)i | + |ui | m 2 2 2 2 i∈Z δ κ 4 2 ∗ |ui | , + |(D u)i | − fi ui + u˙ i ui + 12 2

and

X |i| h 2 2 2 Qθ (t) = θ (δ − ) |u˙ i | + |(Bu)i | − β |(Du)i | m i∈Z i κ +λ |ui |2 + |(D∗ u)i |4 − fi ui . 3

Similarly, as we did in the proof of Lemma 4.2, we obtain that for t ≥ 0, X |i| θ |(Bu(t))i |2 + λ |ui (t)|2 + |u˙ i (t)|2 m i∈Z ≤ max 4, α−1 Mθ (0)e−t + 2α−1 kf k2 ,

(54)

where α > 0 is the same constant chosen in (49). On the other hand, we have

1 |β| λ κ 1

∗ 2 2 2 2 2 2 kDu0 k + ku0 k + Mθ (0) ≤ |u1 k + kBu0 k +

(D u0 ) 2 2 2 2 12 (55) X |i| 2 1 δ 2 2 2 2 + θ |fi | + |u0i | + ku1 k + ku0 k + ku0 k . m 4 2 2 2 i∈Z

T

Since ϕ0 = (u0 , u1 + u0 ) ∈ Or0 , there is a constant K1 > 0 depending on the absorbing ball Or0 , such that max{1, λ + } ku0 k + ku1 k + kBu0 k ≤ K1 . Consequently, kDu0 k ≤ 2 K1 ,

∗ 2

(D u) ≤ 4 K12 .

It follows that there is a uniform constant K2 > 0 depending on K1 , such that X |i| 2 Mθ (0) ≤ K2 + θ |fi | . (56) m i∈Z

θ

Let ϕ (t) =

ϕθi (t) i∈Z

where ϕθi (t)

s |i| = θ ϕi (t). m

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From (54) and (56) we can get ! X |i| 2 θ |fi | , m

θ 2

ϕ (t) ≤ K3 e−t + K4 E

t ≥ 0,

(57)

i∈Z

where K3 and K4 are positive, uniform constants. For any given η > 0, there exists a T0 (η) > 0 such that K3 e−t ≤

η , 2

for all t ≥ T0 (η).

There exists a constant m > 0 such that X η K4 |fi |2 ≤ . 2

(58)

(59)

|i|≥m

As an absorbing set, there is a constant T1 = T1 (Or0 ) > 0 such that S (t) Or0 ⊂ Or0 ,

for all t ≥ T1 .

(60)

Let K(η) = 2m and T (η) = T0 (η) + T1 . Then by (57)–(60), it follows that (52) holds. Theorem 4.1. Under the condition (38) for any given f ∈ `2 , there exists a global attractor in E for the solution semiflow {S(t)}t≥0 generated by the lattice differential equation (36). Proof. We now prove that with (38) and (40) satisfied, the solution semiflow {S (t)}t≥0 generated by (41) is asymptotically compact in E. Given a bounded sequence {ϕn } ⊂ E and tn → ∞, such that kϕn kE ≤ r, n ≥ 1. Let O = Or0 be the absorbing ball shown in Lemma 4.2. Then there exists a constant Tr > 0 such that S (t) {ϕn } ⊂ O,

for all t ≥ Tr .

Hence, there is N1 (r) > 0 such that whenever n ≥ N1 (r) one has tn ≥ Tr , so that S (tn ) {ϕn } ⊂ O,

for n ≥ N1 (r).

(61)

Since a bounded set in a Hilbert space E is weakly precompact, there exists a subsequence of {S (tn ) ϕn }, which is relabeled as the same, such that S (tn ) ϕn → ϕˆ ∈ E weakly, as n → ∞.

(62)

Here we show that S (tn ) ϕn also converges to ϕˆ strongly in E. For any η > 0, by Lemma 4.3 and (61), there exist constants K(η), T (η) > 0 such that X η2 2 |(S (t + Tr ) ϕn )i |E ≤ , for t ≥ T (η), 4 |i|≥K(η)

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where | · |E is defined by (52). Since tn → ∞, there exists N2 (η) such that tn ≥ Tr + T (η) for n ≥ N2 (η). Thus, X X η2 2 2 |(S (tn ) ϕn )i |E = |(S (tn − Tr ) S (Tr ) ϕn )i |E ≤ . (63) 4 |i|≥K(η)

|i|≥K(η)

ˆ On the other hand, since ϕˆ ∈ E, there is a constant K(η) > 0 such that X η2 2 |ϕî |E ≤ . 4

(64)

ˆ |i|≥K(η)

n o ˆ Let K0 (η) = max K(η), K(η) . The weak convergence (62) implies that as n → ∞, (S (tn ) ϕn )i → ϕî ,

for each i with |i| ≤ K0 (η).

Thus there exists N3 (η) > 0 such that X η2 2 |(S (tn ) ϕn )i − ϕî |E ≤ , 2

(65)

for all n ≥ N3 (η).

(66)

|i|≤K0 (η)

Finally, for n ≥ N (η) = max {N1 (r), N2 (η), N3 (η)}, we have X |(S (tn ) ϕn )i − ϕî |2E kS (tn ) ϕn − ϕk ˆ 2E ≤ |i|≤K0 (η)

X

+

2

|(S (tn ) ϕn )i − ϕî |E

|i|>K0 (η)



2

≤

η +2 2

X

2

|(S (tn ) ϕn )i kE +

X

|i|>K0 (η)

|i|>K0 (η)

2

≤η ,

2



(67)

|ϕî |E 

because of (63), (64) and (66). This shows that the solution semiflow {S (t)}t≥0 is asymptotically compact in E. According to Theorem 23.12 in Sell and You, 10 based upon the fact that S (t) has an absorbing set in E and is asymptotically compact in E, there exists a global attractor A in E for the solution semiflow S (t) generated by (41). Note that the original solution semiflow S(t) generated by the lattice differential equation (36) is relative to the semiflow S (t) by S(t) = J− S (t) J , where J =

I

0 I

t ≥ 0,

on E0 = `2 × `2 .

Hence it follows that A = J− A

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is a global attractor in E0 for the semiflow S(t). However, E0 and E are the same Hilbert space with the equivalent norms. So, we have proved that there exists a global attractor A of the semiflow S(t) in E. Furthermore, one can prove that the global attractor A of the solution semiflow S(t) is upper semicontinuous in the sense that lim dE (An , A) = 0

n→∞

with respect to the Hausdorff distance dE in E, where An is the global attractor for the solution semiflow Sn (t) generated by the approximate (2n + 1)-dimensional ODE system which consists of the ith lattice equations, |i| ≤ n, with the cyclic cut-off terms. The details are omitted. References 1. P. W. Bates, K. Lu, and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11(1), 143–153 (2001). 2. J. Boussinesq, Theorie des ondes et des remous qui se propagent le long dun canal, J. Math. Pures Appl., Ser. 2, 17, 55–108 (1872). 3. F. Chen, B. Guo, and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147(2), 231–241 (1998). 4. S.-N. Chow, R. Conti, R. Johnson, J. Mallet-Paret, and R. Nussbaum, Dynamical systems, Lecture Notes in Mathematics, Vol. 1822, Springer-Verlag, Berlin, 2003. Lectures from the C.I.M.E. Summer School held in Cetraro, June 19–26, 2000. 5. P. A. Clarkson, New exact solutions of the Boussinesq equation, European J. Appl. Math., 1(3), 279–300 (1990). 6. R. Hirota, Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique, J. Phys. Soc. Japan, 55(7), 2137–2150 (1986). 7. Y. Li and Q. Chen, Finite-dimensional global attractor for dissipative SchrodingerBoussinesq equations, J. Math. Anal. Appl., 205(1), 107–132 (1997). 8. J. W. Miles, Solitary waves, in: Annual review of fluid mechanics, Vol. 12, pages 11–43. Annual Reviews, Palo Alto, Calif., 1980. 9. A. Nakamura, Exact solitary wave solutions of the spherical Boussinesq equation, J. Phys. Soc. Japan, 54(11), 4111–4114 (1985). 10. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Vol. 143, Springer-Verlag, New York, 2002. 11. B. Straughan, Global nonexistence of solutions to some Boussinesq type equations, J. Math. Phys. Sci., 26(2), 155–164 (1992). 12. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, second edition, 1997. 13. V. Varlamov, Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. Math. Math. Sci., 22(1), 131–145 (1999). 14. Y. You, Nonlinear exponential stabilization of Boussinesq equations, in: Analysis and Optimization of Systems (Antibes, 1990), Lecture Notes in Control and Inform. Sci., Vol. 144, pages 642–651. Springer, Berlin, 1990. 15. S. Zhou, Attractors for second-order lattice dynamical systems, J. Differential Equations, 179(2), 605–624 (2002).

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Chap39-YuanXiaoping

RECENT PROGRESS ON NONLINEAR WAVE EQUATIONS VIA KAM THEORY∗

XIAOPING YUAN School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA Email: [email protected]

In this paper the present author gives out his some works on nonlinear wave equation using KAM theory. In Section 1, some backgrounds are given out for solving the nonlinear wave (NLW) equation via KAM theory. In Section 2, Theorem 2.1 shows that there are many invariant tori for the NLW equation with any non-vanishing prescribed potential. In Section 3, Theorems 3.1 and 3.2 show that there are many invariant tori of any dimension for the completely resonant NLW equation. This answers an open problem proposed by several authors. Keywords: KAM theory, invariant tori, nonlinear wave equation, quasi-periodic solution, completely resonant. 2000 Mathematics Subject Classification: Primary 37K55; Secondary 35L70.

1. Introduction The existence of solutions, periodic in time, for non-linear wave (NLW) equations has been studied by many authors. A wide variety of methods such as bifurcation theory and variational techniques have been brought on this problem. See Ref. 11 and the references therein, for example. There are, however, relatively less methods to find the quasi-periodic solutions of NLW or other PDE’s. The KAM theory is a very powerful tool in order to construct families of quasi-periodic solutions, which are on an invariant manifold, for some nearly integrable Hamiltonian systems of finite many degrees of freedom. In the 1980’s,the celebrated KAM theory has been successfully extended to infinitely dimensional Hamiltonian systems of short range so as to deal with certain class of Hamiltonian networks of weakly coupled oscillators. Vittot & Bellissard, 27 Frohlich, Spencer & Wayne 15 showed that there are plenty of almost periodic solutions for some weakly coupled oscillators of short range. In Ref. 30, it was also shown that there are plenty of quasi-periodic solutions for some weakly coupled oscillators of short range. ∗ Supported

by National Natural Science Foundation of China (10231020). 387

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Because of the restriction of short range, those results obtained in Refs. 27, 15 do not apply to PDE’s. In the 1980-90’s, the KAM theory has been significantly generalized, by Kuksin, 17,18,19 to infinitely dimensional Hamiltonian systems without being of short range so as to show that there is quasi-periodic solution for some class of partial differential equations. Also see P¨ oschel. 24 Let us focus our attention to the following nonlinear wave equation utt − uxx + V (x)u + u3 + h.o.t. = 0,

(1)

subject to Dirichlet and periodic boundary conditions on the space variable x. 1. Dirichlet boundary condition. In 1990, Wayne 28 obtained the time-quasiperiodic solutions of (1), when the potential V is lying on the outside of the set of some “bad” potentials. In Ref. 28, the set of all potentials is given some Gaussian measure and then the set of “bad” potentials is of small measure. Kuksin 17 assumed the potential V depends on n-parameters, namely, V = V (x; a1 , ..., an ), and showed that there are many quasi-periodic solutions of (1) for “most” (in the sense of Lebesgue measure) parameters a’s. However, their results exclude the constant-value potential V (x) ≡ m ∈ R+ , in particular, V (x) ≡ 0. When the potential V is constant, the parameters required can be extracted from the nonlinear term u3 . In order to use the KAM theorem, it is necessary to assume that there are some parameters in the Hamiltonian corresponding to (1). When V (x) ≡ m > 0, these parameters can be extracted from the nonlinear term u3 by Birkhoff normal form 25 , or by regarding (1.1) as a perturbation of sine-Gordon/sinh-Gordon equation 4 . And it was then shown that, for a prescribed potential V (x) ≡ m > 0, there are many elliptic invariant tori which are the closure of some quasi-periodic solutions of (1). By Remark 7 in Ref. 25, the same result holds also true for the parameter values −1 < m < 0. When m ∈ (−∞, −1) \ Z, it is shown in Ref. 29 that there are many invariant tori for (1). In this case, the tori are partially hyperbolic and partially elliptic. 2. periodic boundary condition. In this case, the eigenvalues of the linear d2 operator − dx 2 + V (x) are double (at least, asymptotically double). This results in some additional difficulties in applying KAM technique since the normal frequencies are double. According to our knowledge, the difficulty arising from the multiple normal frequencies (including double ones) was overcome in Ref. 9 in the year of 1969 when the multiplicity is bounded, although Hamiltonian systems are not considered. A key point is to bound the inverse of some matrix by requiring the determinant of the matrix is nonzero. Using Lyapunov-Schmidt decomposition and Newton’s iteration, Craig and Wayne 12 showed that for an open dense set of V (x) there exist time periodic solutions of (1) subject to periodic (also Dirichlet) boundary condition. (The equation considered by them contains more general form than (1). By developing Craig-Wayne’s method, in 1994, Bourgain 7 showed there are many quasi-periodic solutions of (1) for “most” parameters σ ∈ Rn where V = V (x; σ). In 2000, a similar result was obtained by KAM technique in Ref. 13. When the potential V ≡ m 6= 0, the existence of the quasi-periodic solutions was also obtained

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389

in Ref. 5 via the renormalization group method. 2. Nonlinear wave equation with a prescribed potential However, from the works mentioned above one does not know whether there is any invariant tori for prescribed (not random) non-constant-value potential V (x). Recently, the present author has shown that there are many invariant tori for any prescribed non-zero potential V (x) such as sin x and cos x. To give the statement of our results, we need to introduce some notations. We study equation (1) as an infinitely dimensional Hamiltonian system. Following P¨ oschel, 25 the phase space one may take, for example, the product of the usual Sobolev spaces W = H01 ([0, π])× L2 ([0, π]) with coordinates u and v = ut . The Hamiltonian is then H=

1 1 1 hv, vi + hAu, ui + u4 2 2 4

where A = d2 /dx2 − V (x) and h·, ·i denotes the usual scalar product in L2 . The Hamiltonian equation of motions are ∂H ∂H = v, −vt = = Au + u3 . ∂v ∂u Our aim is to construct time-quasi-periodic solutions of small amplitude. Such quasi-periodic solutions can be written in the form ut =

u(t, x) = U (ω1 t, · · · , ωn t, x), where ω1 , · · · , ωn are rationally independent real numbers which are called the basic frequency of u, and U is an analytic function of period 2π in the first n arguments. Thus, u admits a Fourier series expansion X √ u(t, x) = e −1hk,ωit Uk (x), k∈Zn

P

where hk, ωi = j kj ωj and Uk ∈ L2 [0, π] with Uk (0) = Uk (π). Since the quasi-periodic solutions to be constructed are of small amplitude, Equation (1) may be considered as the linear equation utt = uxx − V (x)u with a small nonlinear perturbation u3 . Let φj (x) and λj (j = 1, 2, ...) be the eigenfunctions and eigenvalues of the Sturm-Liouville problem −Ay = λy subject to Dirichlet boundary conditions y(0) = y(π) = 0, respectively. Then every solution of the linear system is the superposition of their harmonic oscillations and of the form X p u(t, x) = qj (t)φj (x), qj (t) = yj cos( λj t + φ0j ) j≥1

with amplitude yj ≥ 0 and initial phase φ0j . The solution u(t, x) is periodic, quasiperiodic or almost periodic depending on whether one, finitely many or infinitely many modes are excited, respectively. In particular, for the choice Nd = {j1 , j2 , · · · , jd } ⊂ N,

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of finitely many modes there is an invariant 2d-dimensional linear subspace ENd that is completely foliated into rational tori with frequencies λj1 , · · · , λjd : ENd = {(u, v) = (qj1 φj1 + · · · + qjd φjd , q˙j1 φj1 + · · · + q˙jd φjd )} [ = Tj (y), y∈P¯d

where Pd = {y ∈ Rd : yj > 0

for 1 ≤ j ≤ d} is the positive quadrant in Rd and

2 TNd (y) = {(u, v) : qj2k + λ−2 jk q˙jk = yk ,

for

1 ≤ k ≤ d}.

Upon restoring the nonlinearity u3 the invariant manifold ENd with their quasiperiodic solutions will not persist in their entirety due to resonance among the modes and the strong perturbing effect of u3 for large amplitudes. In a sufficiently small neighborhood of the origin, however, there does persist a large Cantor subfamily of rotational d-tori which are only slightly deformed. More exactly, we have the following theorem: Theorem R2.1. (Ref. 30) Assume that V (x) is sufficiently smooth in the interval π [0, π], and 0 V (x) dx 6= 0. Let K and N be positive constants large enough. Let Nd = {ip ∈ N : p = 1, 2, · · · , d} with min Nd > N K, max Nd ≤ C0 dN K, and K1 ≤ |ip − iq | ≤ K2 ,

for

p 6= q,

where C0 > 1 is an absolute constant and K1 , K2 , positive constants large enough, depending on K instead of N . Then, for given compact set C ∗ in Pd with positive Lebesgue measure, there is a set C ⊂ C ∗ with meas C > 0, a family of d-tori [ TNd (C) = TNd (y) ⊂ ENd y∈C

over C, and a Lipschitz continuous embedding Φ : TNd [C] ,→ H01 ([0, π]) × L2 ([0, π]) = W, which is a higher order perturbation of the inclusion map Φ0 : ENd ,→ W restricted to TNd [C], such that the restriction of Φ to each TNd (y) in the family is an embedding of a rotational invariant d-torus for the nonlinear equation (1.1). Rπ Basic idea of the proof. It is observed that when 0 V (x) dx 6= 0 the eigenvalues λi ’s satisfy p p p p | λi ± λj ± λk ± λl | ≥ Cm min(i, j, k, l)−1 , if min(i, j, k, l) 1,

unless trivial relations like

p p p p λi − λi + λk − λk ,

where Cm is a constant depending on m. This estimate implies that in the neighborhood of the origin the equation (1) can be put to the Borkhoff normal form 6

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up to terms of the fourth order. Then the application of KAM theorem for PDEs implies that there are many invariant tori for (1) in the neighborhood of the origin. Rπ Remark 2.1. The assumption 0 V (x) dx 6= 0 is not essential. One has p c1 c2 cn 1 λj = j + + 2 +···+ n +O , j j j j n+1 Rπ 1 where cj ’s are some constants depending on V , in particular, c1 = − 2π V (x) dx. 0 Rπ Then the assumption 0 V (x) dx 6= 0 is equivalent to c1 6= 0. The assumption c1 6= 0 is used just only in making Birkhoff normal form. By overcoming more technical trouble one can still get the normal form true under conditions R π c1 = 0, · · · , ck−1 = 0 and ck 6= 0 for some 1 ≤ k ≤ n. Therefore the assumption 0 V (x) dx 6= 0 can be nearly replaced by V (x) 6= 0 in Theorem 2.1. Remark 2.2. Theorem 2.1 still holds true for the following equation X utt = uxx − V (x)u ± u3 + ak u2k+1 m≥k≥2

where m is a positive integer and ak ’s are some real numbers. Remark 2.3. The method in proving Theorem 2.1 can be applied to NLS equation: √ −1ut − uxx + V (x)u ± u3 = 0 subject to Dirichlet boundary conditions. Remark 2.4. If λ1 > 0, then the obtained invariant tori are elliptic. If λ1 < 0, then the tori are hyperbolic-elliptic. Remark 2.5. We can give the measure estimate of the set C: meas C ≥ meas C ∗ · (1 − O(1/13 )). 3. Completely resonant nonlinear wave equation Naturally one can ask: Is there any invariant torus for (1) when V (x) ≡ 0? In this case, the equation is called completely resonant by P o ¨schel. 25 This problem remains open for a relatively long time, which was proposed by many authors, such as P¨ oschel, 25 Craig and Wayne, 12 Kuksin, 18 and Marmi and Yoccoz. 23 The present author answers this question: Theorem 3.1. (Ref. 31) Assume v(x) ≡ 0. For any d ∈ N, the equation (1) subject to the periodic boundary condition possesses many d + 1-dimensional invariant tori in the neighborhood of the equilibrium u ≡ 0. The motions on the tori are quasiperiodic.

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Basic idea of the proof. Assume V (x) ≡ 0. Let u0 (t) be a non-zero solution of the equation u ¨0 + u30 = 0. We will construct the invariant tori or quasi-periodic solutions in the neighborhood of the solution u0 (t) which is uniform in space and periodic in time. To that end, inserting u = u0 + u into (1) we get utt − uxx + 3u20 (t)u + · (h.o.t.) = 0,

x ∈ S 1.

(2)

In considerably rough speaking, by the averaging method we reduce this equation to c2 (0)u + · (h.o.t.) = 0, x ∈ S 1 , utt − uxx + 3u (3) 0 R c2 (0) = 1 2π u2 (t) dt 6= 0. Then we construct the invariant tori or quasiwhere u 0 0 2π 0 periodic solutions of (3) by advantage of c u2 (0) 6= 0. At this time, we should deal 0

with (3) by the same way as in Ref. 25. Unfortunately, one of the frequencies of the Hamiltonian corresponding to (3) is zero (see (6) below). This causes the “integrable” part of the Hamiltonian serious degenerate, incurring great expanse in using KAM technique. Firstly one can easily find the periodic solution u0 (t) of u ¨0 + u30 = 0 with its c2 (0) 6= 0. Then consider a family of Hamiltonian frequency ω and show that u 0 functions p 3u2 (t) Hn = λn zn z¯n + √0 (zn + z¯n )2 , λn 6= 0, n ∈ N = {1, 2, ...}. (4) 4 λn Notice that their equation of motion is linear. By the reducing theory from KAM theory 9,14 we reduce (4) to p 3 2 Hn = µn zn z¯n , µn = λn + ω + O(ω 23/9 /n), n ∈ N, (5) nπ

where λn is the eigenvalues of the Sturm-Liouville problem −y 00 = λy, x ∈ S 1 . At the same time by the Floquet theory 22 one can reduce the Hamiltonian H0 =

1 2 λ0 2 3u20 (t) 2 y + x0 + x0 , 2 0 2 2

λ0 = 0,

(6)

to 1 c0 ω 2 p 2 , (7) 2 where c0 is a constant. In order to exclude the multiplicity of the eigenvalues λn , one can find a solution which is even in the space variable x. That is, one can write P u(t, x) = n≥0 xn (t) cos nx. From this we get a Hamiltonian corresponding to (2) which reads as 1X 2 3 H= yn + λn x2n + u20 (t)x2n + G3 + 2 G4 , (8) 2 2 H0 (q, p) =

n≥0

where G3 (and G4 , resp. ) is a polynomial of order 3 (and 4, resp. ) in variables x0 , x1 , .... Introducing the complex variables we re-write (8):

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H = 12 y02 + 32 u20 (ϑ)x20 + +G3 + 2 G4 .

P

n≥1

√

λn zn z¯n +

2 (t) 3u √0 4 λn (zn

+ z¯n )2

Applying (4)-(7) we get a symplectic transformation Ψ such that X 1 ˜4. ˜ 3 + 2 G H := H ◦ Ψ = (c0 ω 2 )p2 + µn zn z¯n + G 2 n>0

(9)

(10)

˜ 3 and G ˜ 4 involve the time t. Let ϑ = ωt be an angle-variable and Notice that G J =Const. be an action-variable. Then (10) can read as X 1 ˜4, ˜ 3 + 2 G (11) H := H ◦ Ψ = (c0 ω 2 )p2 + Jω + µn zn z¯n + G 2 n>0 ˜ 3 and the non-resonant which is autonomous. One can now kill the perturbations G ˜ 4 by Birkhoff normal form. The one gets part of the perturbation G P H = 21 (c0 ω 2 )p2 + 2 c1 q 4 + 3 O(q 5 ) + n6=0 µn zn z¯n + Jω P + j c2 2 q 2 + 3 O(q 3 ) zj z¯j P + i∈Nd ,j∈N 2 c3 + 3 O(q) zi z¯i zj z¯j + small perturbation ,

(12)

After introducing action-angle variables (I0 , φ0 ) corresponding to (q, p), then (12) reads as P 4/3 H = 2/3 c4 I0 + j>0 µj zj z¯j + Jω P P (13) +Γ(I0 , φ0 ) + j>0 Γj (I0 , φ0 )zj z¯j + j∈N,i∈Nd Γij zi z¯i zj z¯j P + j∈N,i∈Nd 2 c5 zi z¯i zj z¯j + small perturbation .

The ci ’s are constants. Using the averaging method we remove the dependence of Γ, Γj and Γij on the angle variable φ0 . After this, we get a Hamiltonian H = H0 +small perturbation, where H0 is integrable and “twist”. The “twist” property can provide the parameters which we need in using KAM technique. Finally, one gets the invariant tori for (1) with V (x) ≡ 0, by making use of KAM theorem. Remark 3.1. Bourgain, 8 Bambusi-Paleari, 10 Berti-Bolle 2,3 and GentileMastropietro-Procesi 16 construct countably many families of periodic solutions for the nonlinear wave equation utt − uxx ± u3 + h.o.t. = 0. See also Refs. 3, 11, 21 and the references therein for the related problems. More recently, Procesi 26 and Baldi 1 constructed quasi-periodic solutions of 2-dimensional frequency and of Lebesgue measure 0 for the completely resonant nonlinear wave equations. Their construction of quasi-periodic solutions is concise and elegant. Theorem 3.2. (Ref. 31) For any d ∈ N, the equation utt − uxx − u3 = 0

(14)

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subject to the Dirichlet boundary condition u(t, 0) = u(t, π) = 0

(15)

possesses many d-dimensional hyperbolic-elliptic invariant tori in the neighborhood of the equilibrium u ≡ 0. The motions on the tori are quasi-periodic. Basic idea of the proof. Let u0 (t, x) ≡ u0 (x) solve ODE uxx + u3 = 0 with b.c. (15). let u = u0 + ˜ u.

(16)

Inserting (16) into (14) we get u ˜ obeys the following equation and b.c. u ˜tt − u ˜xx − 3u20 u ˜ − 3u0u ˜2 − 2 u ˜3 = 0, (17) u ˜(t, 0) = u ˜(t, π) = 0. Rπ Let V (x) = −3u20 . It is easy to verify 0 V (x) dx 6= 0. By the method similar to that of Theorem 2.1 we can show that the existence of invariant tori. References 1. P. Baldi, Quasi-periodic solutions of the equation utt − uxx + u3 = f (u), preprint. 2. M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearity, Preprint. 3. M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Preprint. 4. A. I. Bobenko and S. Kuksin, The nonlinear Klein-Gordon equation on an interval as perturbed sine-Gordon equation, Comm. Math. Helv. 70, 63–112 (1995). 5. J. Bricmont, A. Kupiainen, and A. Schenkel, Renormalization group and the Melnikov Problems for PDE’s, Commun. Math. Phys. 221, 101–140 (2001). 6. G. D. Birkhoff, Dynamical systems, Am. Math. Soc. Colloq. Publ. IX. New York: American Mathematical Society. VIII, 1927. 7. J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Research Notices 11, 475–497 (1994). 8. J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial equations, Chicago Univ. Press, 69–97 (1999). 9. N. N. Bogoljubov, Yu. A. Mitropoliskii, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York (1976) [Russian Original: Naukova Dumka, Kiev, 1969]. 10. D. Bambusi and S. Paleari, Families of periodic orbits for resonant PDE’s, J. Nonlinear Science, 11, 69–87 (2001). 11. H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. AMS 8, 409–426 (1983). 12. W. Craig and C. Wayne, Newton’s method and periodic solutions of nonlinear wave equation, Commun. Pure. Appl. Math., 46, 1409–1501 (1993). 13. L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211, 497–525 (2000). 14. L. H. Eliasson, Reducibility and point spectrum for linear quasi-periodic skewproducts, Proceedings of ICM 1998, Vol. II, Doc. Math. J. DMV, 779–787.

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15. J. Frohlich, T. Spencer, and C. E. Wayne, Localization in Disordered, Nonlinear Dynamical Systems, J. Statistical Physics, 42, 247–274 (1986). 16. G. Gentile, V. Mastropietro, and M. Procesi, Periodic solutions for completely resonant nonlinear wave equations, Comm. Math. Phys., 256, 437–490 (2005). 17. S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math. 1556, Springer-Verlag, New York (1993). 18. S. B. Kuksin, Elements of a qualitative theory of Hamiltonian PDEs, Proceedings of ICM 1998, Vol. II, Doc. Math. J. DMV, 819–829 (1998). 19. S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21, 192–205 (1987). 20. S. B. Kuksin and J. P¨ oschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr¨ odinger equation, Ann. Math., 143, 149–179 (1996). 21. B. V. Lidskij and E. Shulman, Periodic solutions of the equation utt − uxx + u3 = 0, Funct. Anal. Appl., 22, 332–333 (1988). 22. K. R. Meyer and G. R. Hall, Introduction to Hamiltonian dynamical systems and the N -body problem, Applied Math. Sciences, 90, Springer-Verlag, New York (1992). 23. S. Marmi and J.-C. Yoccoz, Some open problems related to small divisors, Lecture Notes in Math. 1784, Springer-Verlag, New York (2002). 24. J. P¨ oschel, A KAM-theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15, 23, 119–148 (1996). 25. J. P¨ oschel, Quasi-periodic solutions for nonlinear wave equation, Commun. Math. Helvetici, 71, 269–296 (1996). 26. M. Procesi, Quasi-periodic solutions for completely resonant nonlinear wave equations in 1D and 2D, Discr. and Cont. Dyn. Syst., 13, 541–552 (2005). 27. M. Vittot and J. Bellissard, Invariant tori for an infinite lattice of coupled classical rotators. Preprint, CPT-Marseille (1985). 28. C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127, 479–528 (1990). 29. X. Yuan, Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation, Int. J. Math. Math. Science, 18, 1111–1136 (2003). 30. X. Yuan, Quasi-periodic solutions of nonlinear wave equations with a prescribed potential, Discr. and Cont. Dyn. Syst., 3, 615–634 (2006). 31. X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Diff. Eqns., 230, 213–274 (2006).

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