I want to find a process $h$ such that $$m(T) = Em(T) + \int_0^T h(t) dW(t), $$
where $m(T) = e^{ \int_0^T t dW(t)}$. Here, $T$ is some positive constant, and $W(t)$ is Brownian motion.
I get $Em(T) = \frac{1}{6}T^3$. But then I am unsure how to continue.
Hint: Use Ito's formula to expand the martingale $e^{-t^3/6}m(t)$, $0\le t\le T$. Having done this set $t=T$ and multiply through by $e^{T^3/6}$.