Risk Measures and Differential Games

Abstract

We study risk measures in relation to stochastic differential games in a Levy-market. We minimize a risk measure to get a min-max problem. The problem is to find an optimal solution for a convex risk measure in zero-sum games with a 3-dimensional controller. To verify a solution we develop a Hamilton-Jacobi-Bellman-Isacs (HJBI) equation and prove it. Moreover we provide a Nash-equilibrium game that includes scenario optimization.These results are illustrated by entropic risk measure and more general cases. Further, a HJBI equation for dynamic risk measures are shown and proven. We extend our convex risk measure model to include stopping control. Last, a theorem for viscosity solutions are shown and proven.