Stochastic differential equations and applications(Mao Xuerong).....pdf

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ABOUT OUR AUTHOR

Contents

General Notation

1 Brownian Motions and Stochastic Integrals

1.1 INTRODUCTION

1.2 BASIC NOTATIONS OF PROBABILITY THEORY

1.3 STOCHASTIC PROCESSES

1.4 BROWNIAN MOTIONS

1.5 STOCHASTIC INTEGRALS

1.6 ITO’S FORMULA

1.7 MOMENT INEQUALITIES

1.8 GRONWALL-TYPE INEQUALITIES

2 Stochastic Dierential Equations

2.1 INTRODUCTION

2.2 STOCHASTIC DIFFERENTIAL EQUATIONS

2.3 EXISTENCE AND UNIQUENESS OF SOLUTIONS

2.4 Lp-ESTIMATES

2.5 ALMOST SURELY ASYMPTOTIC ESTIMATES

2.6 CARATHEODORY’S APPROXIMATE SOLUTIONS

2.7 EULER–MARUYAMA’S APPROXIMATE SOLUTIONS

2.8 SDE AND PDE: FEYNMAN–KAC’S FORMULA

2.9 THE SOLUTIONS AS MARKOV PROCESSES

3 Linear Stochastic Differential Equations

3.1 INTRODUCTION

3.2 STOCHASTIC LIOUVILLE’S FORMULA

STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS Second Edition

ABOUT OUR AUTHOR Professor Xuerong Mao was born in 1957 in the city of Fuzhou in the province of Fujian, Peoples Republic of China. After obtaining a Diploma at the Textile University, Shanghai, he graduated from the Department of Mathematics with a Masters Degree. His first teaching appointment was Lecturer in the Department of Management at Fuzhou University. Arriving in England in 1987, he was awarded a Doctorate by the Mathematics Institute at the University of Warwick and was then a Post- Doctoral Research Fellow at the Science and Research Engineering Council from 1989-1992. Moving to Scotland, he joined the Department of Statistics and Modelling Science, University of Strathclyde, Glasgow as a lecturer in 1992, was promoted to Reader in 1995 and made Professor in 1998 which post he still holds. Professor Mao is well known and highly respected for his mathematical reputation and has published numerous papers in mathematical journals in both China and the West, in addition to writing four published books; Stability of Stochastic Differential Equations with Respect to Semimartingales (1991), Exponential Stability of Stochastic Differential Equations (1994), Stochastic Differential Equations and Applications (First Edition)(1997) and Stochastic Differential Equations with Markovian Switching (2006). Recognition came from the American Biographical Institute in 2000 with the Millennium Gold Medal of Honour, and he has been visiting Guest Professor of several Chinese universities including Huazhong University of Science and Technology in Wuhan. He is a member of the editorial boards of several international journals including Journal of Stochastic Analysis and Applications and Journal of Dynamics of Continuous, Discrete & Impulsive Systems Series B.

STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS Second Edition Xuerong Mao Department of Statistics and Modelling Science University of Strathclyde, Glasgow Horwood Publishing Chichester, UK HORWOOD PUBLISHING LIMITED

International Publishers in Science and Technology Coll House, Westergate, Chichester, West Sussex, PO20 3QL, England First published in 1997 Second Edition published in 2007 COPYRIGHT NOTICE All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the permission of Horwood Publishing Limited, Coll House, Westergate, Chichester, West Sussex, PO20 3QL, England. © Horwood Publishing Limited, 2007. British Library Cataloguing in Publication Data A catalogue record of this book is available from the British Library ISBN: 978-1-904275-34-3 Cover design by Jim Wilkie Printed and bound in the UK by Antony Rowe Limited.

DEDICATION To my parents: Mr Mao Yiming and Mrs Chen Dezhen v

Preface to the Second Edition In this new edition, I have added some material which is particu- larly useful in applications, namely the new Section 9.3 on options and their values and the new Chapter 11 on stochastic delay population systems. In addition, more material has been added to Section 9.2 to include several popular stochastic models in ﬁnance, while the con- cept of the maximal local solution to a stochastic functional diﬀerential equation has been added to Section 5.2 which forms a fundamental theory for our new Chapter 11. During this work, I have beneﬁtted from valuable comments and help from several people, including K.D. Elworthy, G. Gettinby, W. Gurney, D.J. Higham, N. Jacob, P. Kloeden, J. Lam, X. Liao, E. Renshaw, A.M. Stuart, A. Truman, G.G. Yin. I am grateful to them all for their help. I would like to thank the EPSRC/BBSRC, the Royal Society, the London Mathematics Society as well as the Edinburgh Mathematical Society for their ﬁnancial support. Moreover, I should thank my family, in particular, Weihong, for their constant support. Xuerong Mao Glasgow June 2007 vii

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