We prove that for any constant $K\geq1$, the value functions for time homogeneous stochastic differential games in the whole space can be approximated up to a constant over $K$ by value functions whose second-order derivatives are bounded by a constant times $K$.
On the way of proving this result we prove that the value functions for stochastic differential games in domains and in the whole space admit estimates of their Lipschitz constants in a variety of settings.
"On regularity properties and approximations of value functions for stochastic differential games in domains." Ann. Probab. 42 (5) 2161 - 2196, September 2014. https://doi.org/10.1214/13-AOP848