Dynamic Risk Measures and Path-Dependent Second Order PDEs.

Summary We propose new notions of regular solutions and viscosity solutions for path-dependent second order partial differential equations. Making use of the martingale problem approach to path-dependent diffusion processes, we explicitly construct families of time-consistent dynamic risk measures on the set of cadlag paths $I{R}^n$ valued endowed with the Skorokhod topology. These risk measures are shown to have regularity properties. We prove then that these time-consistent dynamic risk measures provide viscosity supersolutions and viscosity subsolutions for path-dependent semi-linear second order partial differential equations.