Suppose that one wants to control a diffusion process $$ dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x $$ in order to optimize a stochastic control problem with value function $$ V_T(u)=\mathbb{E}[f(X^u_T) + \int_0^T g(X_t^u)dt]. $$ Where $f,g$ are bounded smooth functions and $u$ is an admissible control.
Now suppose that one wants to optimize the deterministic control problem: $$ \partial_t x_t^u = \mu(x_t^u,u)dt ; \qquad X_0^u=x $$ with value function $$ v_T(u)=f(X^u_T) + \int_0^T g(X_t^u)dt. $$
For small $\epsilon$, when can one guarantee that the optimal policies of both problems are "close" (in expectation or some other appropriate sense)?
