  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.
