Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

✍ Scribed by Nizar Touzi (auth.)

• Publisher: Springer-Verlag New York
• Year: 2013
• Tongue: English
• Leaves: 218
• Series: Fields Institute Monographs 29
• Edition: 1
• Category: Library

✦ Synopsis

​This book collects some recent developments in stochastic control theory with applications to financial mathematics. We first address standard stochastic 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 stochastic target problems which extends in a nontrivial way the standard stochastic 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 second order 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 stochastic differential equations, and their extensions to the quadratic case.​

✦ Table of Contents

Front Matter....Pages i-x
Introduction....Pages 1-4
Conditional Expectation and Linear Parabolic PDEs....Pages 5-20
Stochastic Control and Dynamic Programming....Pages 21-37
Optimal Stopping and Dynamic Programming....Pages 39-51
Solving Control Problems by Verification....Pages 53-66
Introduction to Viscosity Solutions....Pages 67-88
Dynamic Programming Equation in the Viscosity Sense....Pages 89-99
Stochastic Target Problems....Pages 101-121
Second Order Stochastic Target Problems....Pages 123-147
Backward SDEs and Stochastic Control....Pages 149-164
Quadratic Backward SDEs....Pages 165-188
Probabilistic Numerical Methods for Nonlinear PDEs....Pages 189-199
Introduction to Finite Differences Methods....Pages 201-212
Back Matter....Pages 213-214

✦ Subjects

Quantitative Finance; Probability Theory and Stochastic Processes; Partial Differential Equations; Calculus of Variations and Optimal Control; Optimization