Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following hold, and if yes, is there a monograph where I can find such a result?
$$ \mathbb{P}(X(s) \in A\ \forall s\in[0,t]) = \lim_{n\to\infty}\mathbb{P}(X(kt/n)\in A\ \forall k=1,\ldots,n) $$ Additionally, does the claim stay true if I let the set $A$ vary with $s$? What does the family $(A(s))_{s\ge 0}$ has to satisfy such that this is the case?
I appreciate your help.