Continuous-time Stochastic Control and Optimization with Financial Applications
Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control.
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Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics)
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Edited by B. Rozovski˘ı G. Grimmett Advisory Board D. Dawson D. Geman I. Karatzas F. Kelly Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux
Huyên Pham
Continuous-time Stochastic Control and Optimization with Financial Applications
Huyên Pham Université Paris 7 - Denis Diderot UFR Mathématiques Site Chevaleret, Case 7012 75202 Paris Cedex 13 France [email protected]
Managing Editors Boris Rozovski˘ı Division of Applied Mathematics Brown University 182 George St Providence, RI 02912 USA [email protected]
Geoffrey Grimmett Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK [email protected]
ISSN 0172-4568 ISBN 978-3-540-89499-5 e-ISBN 978-3-540-89500-8 DOI 10.1007/978-3-540-89500-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926070 Mathematics Subject Classification (2000): 93E20, 91B28, 49L20, 49L25, 60H30 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Chˆau, Hugo and Antoine
Preface
Dynamic stochastic optimization is the study of dynamical systems subject to random perturbations, and which can be controlled in order to optimize some performance criterion. It arises in decision-making problems under uncertainty, and finds numerous and various applications in economics, management and finance. Historically handled with Bellman’s and Pontryagin’s optimality principles, the research on control theory has considerably developed over recent years, inspired in particular by problems emerging from mathematical finance. The dynamic programming principle (DPP) to a stochastic control problem for Markov processes in continuous-time leads to a nonlinear partial differential equation (PDE), called the Hamilton-JacobiBellman (HJB) equation, satisfied by the value function. The global approach for studying stochastic control problems by the Bellman DPP has a suitable framework in viscosity solutions, which have become popular in mathematical finance: this allows us to go beyond
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