Application of martingale representation theorem

Question

I am reading a proof that uses the following fact without proof (a bit strange):

Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq 0}$. Also, let $\phi$ be a bounded, continuously differentiable function with bounded derivative $\phi'$. Let $T>0$ be constant.

Then it claims that, by martingale representation theorem, there exists a predictable process $\beta$ such that $\mathbb{E} \int_0^{\infty} \beta^2_u \,du ) < +\infty$ and a constant $c$ such that $$\phi(W_T)= c + \int_0^{\infty} \beta_u \,dW_u.$$

But I haven't seen the fact the a bounded $C^1$ function of a Brownian motion is a local martingale, which is a sufficient condition to apply the martingale representation theorem.

Answer 1

Define a martingale by

$$M_t := \begin{cases} \mathbb{E}(\phi(W_T) \mid \mathcal{F}_t), & t \leq T, \\ \phi(W_T), & t>T. \end{cases}$$

Then, by the martingale representation theorem, there exists a representation of the form

$$M_t = c+ \int_0^t \beta_s \, dW_s.$$

For $t=T$, this proves the claim.

Remark: One possibility to prove the mentioned martingale representation theorem is proving the following representation theorem:

Let $(B_t)_{t \geq 0}$ be a one-dimensional Brownian motion and $(\mathcal{F}_t))_{t \geq 0}$ an admissible right-continuous filtration. If $X \in L^2$ is $\mathcal{F}_T$-measurable, then there exists a progressively measurable process $\beta$ and a constant $c$ such that $$X = c + \int_0^T \beta(s) \, dB_s.$$

If you know this theorem, you can apply it directly to the given random variable $X = \phi(W_T)$.