Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

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 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 stochastic differential equations, and their extensions to the quadratic case.

Provides a self-contained presentation of the recent developments in Stochastic target problems which cannot be found in any other monograph

1. Introduction.- 2. Conditional Expectation and Linear Parabolic PDEs.- 3. Stochastic Control and Dynamic Programming.- 4. Optimal Stopping and Dynamic Programming.- 5. Solving Control Problems by Verification.- 6. Introduction to Viscosity Solutions.- 7. Dynamic Programming Equation in the Viscosity Sense.- 8. Stochastic Target Problems.- 9. Second Order Stochastic Target Problems.- 10. Backward SDEs and Stochastic Control.- 11. Quadratic Backward SDEs.- 12. Probabilistic Numerical Methods for Nonlinear PDEs.- 13. Introduction to Finite Differences Methods.- References.