Stochastic singular control: Existence, characterization and approximation of solutions in cost minimization problems and games
A talk in the Bielefeld Stochastic Afternoon series by
Jodi Dianetti
| Abstract: | $$\textbf{Bielefeld Stochastic Afternoon - Math Finance Session}$$
For a class of optimal stochastic control problems with singular controls, we characterize the optimal control as the unique solution to a related Skorokhod reflection problem. We prove that the optimal control only acts when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. We next consider problems concerning existence and approximation of equilibria in N-player stochastic games and mean field games of singular control. In a not necessarily Markovian setting, we establish the existence of Nash and mean field equilibria for games with submodular costs via Tarski's fixed point theorem. This approach allows to prove that there exist minimal and maximal equilibria which can be obtained through a simple learning procedure based on the iteration of the best-response-map. Finally, we analyse stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and an ergodic performance criterion. We prove existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations.
This talk is based on joint works with Giorgio Ferrari, Markus Fischer and Max Nendel. Within the CRC this talk is associated to the project(s): C3, C4, C5 |