If we have a particle whose position $X(t)$ is stochastically described by:
$$X(t+\Delta t)=X(t)+f(X(t),t)\Delta t+g(X(t),t)\sqrt{\Delta t}\zeta$$
Where $\zeta \sim N(0,1)$. What is we want to consider the particle's movement in terms of $X(s)$ and $X(s+\Delta s)$. Is this rigorous enough?
$$X(t)=X(t+\Delta t)-f(X(t),t)\Delta t-g(X(t),t)\sqrt{\Delta t}\zeta.$$ Then let $t+\Delta t= s$ and $t=s-\Delta s$, $\Rightarrow \Delta t=\Delta s$
Giving $$X(s-\Delta s)=X(s)-f(X(s-\Delta s),s-\Delta s)\Delta s-g(X(s-\Delta s),s-\Delta s)\sqrt{\Delta s}\zeta$$.
Giving the expectation functions, $$E[X(s)-y|X(s-\Delta s)=y]=E[f(y,s-\Delta s)\Delta s+g(y,s-\Delta s)\sqrt{\Delta s}\zeta]$$ $$=f(y,s-\Delta s)\Delta s$$?