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

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