Asymptotic $C^{1,\gamma }$-regularity for value functions to uniformly elliptic dynamic programming principles

In this paper we prove an asymptotic $C^{1,\gamma }$-estimate for value functions of stochastic processes related to uniformly elliptic dynamic programming principles. As an application, this allows us to pass to the limit with a discrete gradient and then to obtain a $C^{1,\gamma }$-result for the corresponding limit PDE.