Viscosity solutions for fully nonlinear second order equations with applications to stochastic differential games

In this paper we relate a recent result in the theory of viscosity solutions of second order partial differential equations and we sketch an application of this result to stochastic control theory. The result related is a weak maximum principle for viscosity solutions of fully nonlinear degenerate elliptic partial differential equations or FNDEPDEs for short. As an application of this result we prove existence of value of stochastic differential games under the Isaac's condition. We just outline the proof since a rigorous treatment would require dealing with far too many technical details for the space available and would also tend to obscure the basic ideas. The maximum principal we relate is taken from [3] where a complete detailed proof may be found. This is the first direct proof of such a result and the first proof that includes nonconvex and nonconcave FNDEPDEs. It is this last fact which is crucial to the application we make. Proofs for convex or concave FNDEPDEs may be found in [5] and [6]. In fact, what is actually proved in these papers is the equivalence of the viscosity solution and the value function of an associated stochastic optimal control problem. The maximum principle then follows from stochastic optimal control theory. As such, these earlier results are clearly not applicable to stochastic differential games. Our proof of the existence of value is similar in spirit to those proofs given for deterministic differential games which are based on the maximum principle in [1], i.e. on the uniqueness of viscosity solutions of first order equations. Unlike the situation in deterministic differential games, the existence of value for stochastic differential games, under the Isaacs' condition, was previously unknown. This application represents a new result in stochastic control theory.