deterministic expression of stochastic integral

Question

Let $(M_t)$ be a non-negative martingale on a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_n \} , \mathbb{P})$. Let $dM_t = M_t dW_t$. How can we write the following \begin{equation} \mathbb{E} [\sqrt{M_T}| \mathcal{F}_t], \end{equation} where $0 \leq t \leq T$, in terms of a deterministic function $F(t,T,M_t)$?

Answer 1

Hint: Solve the stochastic differential equation $$dM_t = M_t \, dW_t$$ in order to find an explicit formula for $M_t$ in terms of $t$ and $W_t$.