On representation and regularity of viscosity solutions to degenerate Isaacs equations and certain nonconvex Hessian equations

We study the smoothness of the upper and lower value functions of stochastic differential games in the framework of time-homogeneous (possibly degenerate) diffusion processes in a domain, under the assumption that the diffusion, drift and discount coefficients are all independent of the spatial variables. Under suitable conditions (see Assumptions 2.1 and 2.2), we obtain the optimal local Lipschitz continuity of the value functions, provided that the running and terminal payoffs are globally Lipschitz. As applications, we obtain the stochastic representation and optimal interior $C^{0,1}$-regularity of the unique viscosity solution to the Dirichlet problem for certain degenerate elliptic, nonconvex Hessian equations in suitable domains, with Lipschitz boundary data.