Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Stochastic 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. Stochastic Target Problems.- 8. Second Order Stochastic Target Problems.- 9. Backward SDEs and Stochastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References.
¿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.¿
¿Provides a self-contained presentation of the recent developments in Stochastic target problems which cannot be found in any other monograph
Approaches quadratic backward stochastic differential equations following the point of view of Tevzadze and presented in a way to maximize the ease of understanding
Contains relevant examples from finance, including the Nash equilibrium example¿