Fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ that supports a cadlag Markov process $(X_t)_{t\geq 0}$. Let $(\xi_t)_{t\geq 0}$ be a cadlag (right-continuous with left limit) process and adapted to the natural filtration of $X$, i.e. $\xi_t\in \mathcal{F}^{X}_t$ for all $t\geq 0$.
My question is: Then under what extra conditions on $X$ do we have
i): $\xi_t=F(t\mapsto X_t)$ where $F$ is a functional of the whole path of $X$;
ii): $\xi_t=f(t,X_t)$;
iii): $\xi_t=g(X_t)$.