I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. For reference I want to find a stochastic process $X$ such that for a Brownian martingale $M$ we can write $M = X \cdot W$ plus maybe a constant. Where the dot stands for stochastic integral notation.
For example we could define $M_t = \mathbb{E}\left[ e^{W_T} | \mathcal{F}_t \right]$ for a fixed time $T$ then it is a martingale for the filtration generated by $W$. But how could I construct a process $X$ such that $M = X \cdot W$? I pretty much don't have a clue how to proceed so thanks for any help.
Best Answer
Since $$\mathbb E[e^{W_T}|\mathcal F_t]=\mathbb E[e^{W_T-W_t}e^{W_t}|\mathcal F_t]=e^{W_t}\mathbb E[e^{W_T-W_t}|\mathcal F_t]=e^{W_t}\mathbb E[e^{W_T-W_t}]=e^{W_t+\frac{1}{2}(T-t)}\,,$$ an application of Itô's lemma gives $\mathbb E[e^{W_T}|\mathcal F_t]=\int_{0}^{t}e^{W_u+\frac{1}{2}(T-u)}\mathrm dW_u\ (\text{for }0\leqslant t\leqslant T)$, which is of the desired form and $e^{W_t+\frac{1}{2}(T-t)}$ is the "$X$" process you seek.