Stochastic Optimal Control in Infinite Dimension: Dynamic Programming and HJB Equations

by Giorgio Fabbri, Fausto Gozzi, Andrzej Swiech, Marco Fuhrman, Gianmario Tessitore

Overview

Providing an introduction to shastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated shastic optimal control problems. It features a general introduction to optimal shastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional shastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of shastic PDEs,and in PDEs in infinite dimension. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to shastic optimal control problems in finite dimension, and the basics of shastic analysis and shastic equations in infinite-dimensional spaces.

Product Details

ISBN-13:	9783319530666
Publisher:	Springer International Publishing
Publication date:	06/24/2017
Series:	Probability Theory and Stochastic Modelling , #82
Edition description:	1st ed. 2017
Pages:	916
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Giorgio Fabbri is a CNRS Researcher at the Aix-Marseille School of Economics, Marseille, France. He works on optimal control of deterministic and shastic systems, notably in infinite dimensions, with applications to economics. He has also published various papers in several economic areas, in particular in growth theory and development economics.

Fausto Gozzi is a Full Professor of Mathematics for Economics and Finance at Luiss University, Roma, Italy. His main research field is the optimal control of finite and infinite-dimensional systems and its economic and financial applications. He is the author of many papers in various subjects areas, from Mathematics, to Economics and Finance.

Andrzej Swiech is a Full Professor at the School of Mathematics, Georgia Institute of Technology, Atlanta, USA. He received Ph.D. from UCSB in 1993. His main research interests are in nonlinear PDEs and integro-PDEs, PDEs in infinite dimensional spaces, viscosity solutions, shastic and deterministic optimal control, shastic PDEs, differential games, mean-field games, and the calculus of variations.

*Marco Fuhrman* is a Full Professor of Probability and Mathematical Statistics at the University of Milano, Italy. His main research topics are shastic differential equations in infinite dimensions and backward shastic differential equations for optimal control of shastic processes.

*Gianmario Tessitore* is a Full Professor of Probability and Mathematical Statistics at Milano-Bicocca University. He is the author of several scientific papers on control of shastic differential equations in finite and infinite dimensions. He is, in particular, interested in the applications of backward shastic differential equations in shastic control.

Table of Contents

Preface.- 1.Preliminaries on shastic calculus in infinite dimensions.- 2.Optimal control problems and examples.- 3.Viscosity solutions.- 4.Mild solutions in spaces of continuous functions.- 5.Mild solutions in L2 spaces.- 6.HJB Equations through Backward Shastic Differential Equations (by M. Fuhrman and G. Tessitore).- Appendix A, B, C, D, E.- Bibliography.

Editorial Reviews

This book addresses a comprehensive study of the theory of shastic optimal control when the underlying dynamic evolves as a shastic differential equation in infinite dimension. It contains the most general models appearing in the literature and at the same time provides interesting applications. The book is well written and is mainly addressed to graduate students of engineering and of pure and applied mathematics.” (Hector Jasso, zbMATH 1379.93001, 2018)

From the Publisher