Let $W_t=\int_0^t dW_s$ be a wiener process and let $Y_t = Y_t(W_t)$ be a martingale. By the martingale representation theorem $dY_t = h_t dW_t$ for some unique, predictable process $h_t$. Given that we know that $Y_T = g(W_T)$ for some function $g(x)$ and at some time point $T$, is it possible to find $h_t$ for $t<T$?
Intuitively I had the idea to apply Itô's lemma at the time $T$ to get \begin{equation} dY_T = \frac{\partial g}{\partial x}(W_T) dW_T \end{equation} and identify $h_T = \frac{\partial g}{\partial x}(W_T)$ and then simply let $h_t = E[h_T | \mathcal{F}_t]$. This actually seems to work for the specific problem I'm looking at, but it's not exactly formal and I don't think it would hold as a proof. Is there any validity to this idea? What other methods could be used to find $h_t$?