Thank you very much.

Thanks for the answer. $\tilde f$ should be a function defined on space $D([0,T];R^d)$? Then, the problem is: $f$ is bounded continuous on $R^d$, is it true that $\tilde f(\omega,t):=f(\omega_t)=f(X_t(\omega))$ is bounded an continuous on $D([0,T];R^d)\times[0,T]$?

Thanks again for your answer. You answer is correct. I re-asked a question in a different formulation. But your answer here really helps a lot.

Thanks a lot for that.

Thanks. I changed the problem sightly, so the counterexample you provided does not hold any more.

Thanks. Can you provide a reference? Thank you.

Sorry. I just change $X$ to $R^n$ for simple.