Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions

TL;DR

This paper proves a verification theorem for zero-sum stochastic differential games, establishing conditions for Nash equilibria using a chain rule for non-smooth functions and extending results to degenerate diffusions.

Contribution

It introduces a verification theorem for stochastic differential games based on a chain rule for non-smooth functions, applicable to degenerate diffusions and improving existing stochastic control results.

Findings

Existence of saddle points under Isaacs' condition

Value of the game equals the solution of Bellman-Isaacs equations

Extension to degenerate diffusion cases

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 non-degenerate, 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.