Verification Theorem Related to a Zero Sum Stochastic Differential Game Based on a Chain Rule for Nonsmooth Functions

Abstract. In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs’ condition and the existence of a (what is termed) quasi-strong solution to the Bellman–Isaacs (BI) equations. If the diffusion coefficient of the state equation is nondegenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game; moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly, we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors.