Verification Theorem Related to a Zero Sum Stochastic Differential Game Via Fukushima-Dirichlet Decomposition

We establish a verification theorem, inspired by those existing in stochastic control, to demonstrate how a pair of progressively measurable controls can form a Nash equilibrium in a stochastic zero-sum differential game. Specifically, we suppose that a pathwise-type Isaacs condition is satisfied together with the existence of what is termed a quasi-strong solution to the Bellman-Isaacs (BI)equations. In that case we are able to show that the value of the game is achieved and corresponds exactly to the unique solution of the BI equations. Those have also been applied for improving a well-known verification theorem in stochastic control theory. In so doing, we have implemented new techniques of stochastic calculus via regularizations, developing specific chain rules.