Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

​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 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 shastic differential equations, and their extensions to the quadratic case.​