Let $(M_{t})_{t \geq0}$ be a local martingale such that $M_{0} = 0$, I want to prove the following:
There exists an unique $H$ with $\int _{0}^t (H_{s})^2 ds < \infty$ such that $M_{t} = \int _{0}^{t}H dW $
I have already proven uniqueness and the fact that $M_{t}$ always has a continuous version (such that the above representation is always well-defined). However, I'm still struggling with finding an explicit form for $H$.
Obviously, we need to use a localizing sequence $(\tau _{n})_{n \geq 1}$ for $M_{t}$, as then we can use the representation theorem for regular martingales to write for all $n \geq 1$: $M_{t \wedge \tau_{n}} = \int _{0}^{t}H_{n} dW$, where $H_{n}$ has the same properties as $H$ above. My question is then, how to use this sequence to construct an explicit form for $H$?