Wendell H. Fleming Stochastic Analysis, Control, Optimization and Applications A Volume in Honor of W.H. Fleming William M. McEneaney, G. George Yin, and Qing Zhang Editors Springer-Science+Business Media, LLC William M. McEneaney G. George Yin Department of Mathematics Department of Mathematics North Carolina State University Wayne State University Raleigh, NC 27695 Detroit, MI 48202 Qing Zhang Department of Mathematics University of Georgia Athens, GA 30602 Library of Congress Cataloging-in-Publieatioii Data Stochastic analysis, control, optimization and applications : a volume in honor of W.H. Fleming / William McEneaney, G. George Yin, Qing Zhang, [editors], p. cm. Includes bibliographical references and index. ISBN 978-1-4612-7281-6 ISBN 978-1-4612-1784-8 (eBook) DOI 10.1007/978-1-4612-1784-8 1. Stochastic analysis. 2. Control theory. I. Fleming, Wendell Helms, 1928- . II. McEneaney, William. III. Yin, George, 1954- . IV. Zhang, Qing, 1959- QA274.2.S7718 1999 519.2—dc21 98-45990 CIP AMS Subject Classifications: 90,93 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 1999 Softcover reprint of the hardcover 1st edition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer-Science-+Business Media, LLC, except for brief excerpts in, connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-7281-6 Typeset by the editors in TgC. 987654321 Dedicated to Wendell H. Fleming On the Occasion of His 70th Birthday Contents Preface ............................................................... xi Foreword ........................................................... xiii W.H. Fleming's Curriculum Vitae .................................. xvii Contributors and Addresses ....................................... xxvii PART I. LARGE DEVIATIONS, RISK SENSITIVE AND Hoo CONTROL 1. Representations for Functionals of Hilbert Space Valued Diffusions A. BUDHIRAJA and P. DUPUIS ..................................... 1 2. Risk-Sensitive, Minimax, and Mixed Risk-Neutral/Minimax Control of Markov Decision Processes S.P. CORALUPPI and S.1. MARCUS ................................ 21 3. Partially Observed Control Problems with Multiplicative Cost D. HERNANDEZ-HERNANDEZ ...................................... 41 4. Nonlinear Semigroups for Partially Observed Risk-Sensitive Control and Minimax Games M.R. JAMES ...................................................... 57 5. Nonlinear, Dissipative, Infinite Dimensional Systems M. KOCAN and P. SORAVIA ....................................... 75 6. Singular Limits of Bellman Equations of Ergodic Type Related to Risk-Sensitive Control H. NAGAI ......................................................... 95 7. Game Approach to Risk Sensitive Control for Stochastic Evolution Systems M. NISIO ........................................................ 115 8. On the Solutions of the Equation Arising from the Singular Limit of Some Eigen Problems S.-J. SHEU and A.D. WENTZELL ................................ 135 9. Nonlinear Hoo Controller Design via Viscosity Supersolutions of the Isaacs Equation M. XIAO and T. BA~AR ......................................... 151 viii Contents PART II. PARTIAL DIFFERENTIAL EQUATIONS AND VISCOSITY SOLUTIONS 10. Singularities of Semiconcave Functions in Banach Spaces P. ALBANO and P. CANNARSA .................................. 171 11. Invariant Sets for Controlled Degenerate Diffusions: A Viscosity Solutions Approach M. BARDI and P. GOATIN ...................................... 191 12. Remarks on the Dirichlet Problem for Quasilinear Elliptic and Parabolic Equations G. BARLES, E. Rouy, and P.E. SOUGANIDIS ................... 209 13. A Generalized Hamilton-Jacobi-Bellman Equation for Deterministic Optimal Control Problems L.D. BERKOVITZ ................................................ 223 14. Regular Solutions of Stochastic Burgers Equation P.L. CHOW ..................................................... 237 15. Piecewise-Deterministic Processes and Viscosity Solutions M.H.A. DAVIS and M. FARID ................................... 249 16. Mathematical Approaches to the Problem of Noise-Induced Exit M.V. DAy ...................................................... 269 17. An Approximation Scheme for Evolutive Hamilton-J acobi Equations M. FALCONE and T. GIORGI .................................... 289 18. Homogenization of the Cauchy Problem for Hamilton-Jacobi Equations H. ISHII ......................................................... 305 19. The Critical Exponent for a Stochastic PDE to Hit Zero C. MUELLER and E. PARDOUX .................................. 325 PART III. STOCHASTIC CONTROL, FILTERING AND PARAMETER ESTIMATION 20. Robustness of Zakai's Equation via Feynman-Kac Representations R. ATAR, F. VIENS, and O. ZEITOUNI .......................... 339 21. Estimation of Probability Distributions for Individual Parameters Using Aggregate Population Data H.T. BANKS, B.G. FITZPATRICK, L.K. POTTER, and Y. ZHANG 353 22. Solvable Infinite Time Horizon Stochastic Control Problems in Noncompact Symmetric Spaces T .E. DUNCAN .................................................. 373 23. Exact Finite Dimensional Filters for Exponential Functionals of the State R.J. ELLIOTT and V. KRISHNAMURTHY ......................... 391 Contents ix 24. A Lyapunov Theory of Nonlinear Observers A.J. KRENER ................................................... 409 25. Existence of Optimal Controls for Variance Control H.J. KUSHNER .................................................. 421 26. On Optimal Ergodic Control of Diffusions with Jumps J.-L. MENALDI and M. ROBIN .................................. 439 27. Markov Marginal Problems and Their Applications to Markov Optimal Control T. MIKAMI ..................................................... 457 28. Entropy Inequalities and Entropy Dynamics in Nonlinear Filtering of Diffusion Processes D. OCONE ...................................................... 477 29. Identification for Linear Stochastic Distributed Parameter Systems with Boundary/Point Control B. PASIK-DuNCAN .............................................. 497 30. Monte Carlo Estimation of Diffusion Distributions at Inter-sampling Times C.J. WYPASEK, J.V. BUTERA, and B.G. FITZPATRICK ......... 505 PART IV. MATHEMATICAL FINANCE AND OTHER APPLICATIONS 31. Option Pricing in a Market with Frictions A. BENSOUSSAN and H. JULIEN ................................. 521 32. Pathwise Comparison of Arithmetic Brownian Motions and Log-normal Processes G. FERREYRA and P. SUNDAR .................................. 541 33. Critical Power for Asymptotic Connectivity in Wireless Networks P. GUPTA and P.R. KUMAR .................................... 547 34. Pricing Models with Transaction Fees J .E. HODDER and T. ZARIPHOPOULOU ......................... 567 35. A Verification Theorem in General Equilibrium Model of Asset Prices C.-F. HUANG, M. TAKSAR, and S.H. ZHU ...................... 585 36. Optimal Portfolio Management with Partial Observations and Power Utility Function R. RISHEL ...................................................... 605 37. Hierarchical Production Controls for a Stochastic Manufacturing System with Long-Run Average Cost: Asymptotic Optimality S.P. SETHI and H. ZHANG ...................................... 621 Preface In view of Professor Wendell Fleming's many fundamental contributions, his profound influence on the mathematical and systems theory communi­ ties, his service to the profession, and his dedication to mathematics, we have invited a number of leading experts in the fields of control, optimiza­ tion, and stochastic systems to contribute to this volume in his honor on the occasion of his 70th birthday. These papers focus on various aspects of stochastic analysis, control theory and optimization, and applications. They include authoritative expositions and surveys as well as research papers on recent and important issues. The papers are grouped according to the following four major themes: (1) large deviations, risk sensitive and Hoc control, (2) partial differential equations and viscosity solutions, (3) stochastic control, filtering and parameter esti­ mation, and (4) mathematical finance and other applications. We express our deep gratitude to all of the authors for their invaluable contributions, and to the referees for their careful and timely reviews. We thank Harold Kushner for having graciously agreed to undertake the task of writing the foreword. Particular thanks go to H. Thomas Banks for his help, advice and suggestions during the entire preparation process, as well as for the generous support of the Center for Research in Scientific Computation. The assistance from the Birkhauser professional staff is also greatly appreciated. Raleigh, NC William McEneaney Detroit, MI George Yin Athens, GA Qing Zhang Foreword Wendell Fleming has made fundamental contributions to many exciting ar­ eas in contemporary mathematics and applied mathematics. These include geometric