In this talk, we introduce a notion of set valued PDEs. The set values have been introduced for many applications, such as time inconsistent stochastic optimization problems, multivariate dynamic risk measures, and nonzero sum games with multiple equilibria. One crucial property they enjoy is the dynamic programming principle (DPP). Together with the set valued Itô formula, which is a key component, DPP induces the PDE. In the context of multivariate optimization problems, we introduce the set valued Hamilton-Jacobi-Bellman equations and established its wellposedness. In the standard scalar case, our set valued PDE reduces back to the standard HJB equation.

Our approach is intrinsically connected to the existing theory of surface evolution equations, where a well-known example is mean curvature flows. Roughly speaking, those equations can be viewed as first order set valued ODEs, and we extend them to second order PDEs. Another difference is that, due to different applications, those equations are forward in time (with initial conditions), while we consider backward equations (with terminal conditions). The talk is based on a joint work with Prof. Jianfeng Zhang.