Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE / Edition 1

by Nizar Touzi

Overview

This book collects some recent developments in shastic control theory with applications to financial mathematics. We first address standard shastic control problems from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on the regularity issues and, in particular, on the behavior of the value function near the boundary. We then provide a quick review of the main tools from viscosity solutions which allow to overcome all regularity problems. We next address the class of shastic target problems which extends in a nontrivial way the standard shastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. Namely, the secondorder extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part specializes to an overview of Backward shastic differential equations, and their extensions to the quadratic case.​

Product Details

ISBN-13:	9781461442851
Publisher:	Springer New York
Publication date:	09/27/2012
Series:	Fields Institute Monographs , #29
Edition description:	2013
Pages:	214
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Shastic Control and Dynamic Programming.- 3. Optimal Stopping and Dynamic Programming.- 4. Solving Control Problems by Verification.- 5. Introduction to Viscosity Solutions.- 6. Dynamic Programming Equation in the Viscosity Sense.- 7. Shastic Target Problems.- 8. Second Order Shastic Target Problems.- 9. Backward SDEs and Shastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References.

Editorial Reviews

This is an excellent book on the topic of Shastic Control Problems (SCP). The author transformed his notes for a graduate course at the Field Institute into a volume that will serve also as a good reference in the area. … The author has chosen the framework of diffusions, which makes the exposition more friendly and accessible to a larger audience, in particular for those who want to learn this topic.” (Jaime San Martín, Bulletin of the American Mathematical Society, Vol. 54 (2), April, 2017)

From the Publisher