= Clark-Ocone formula for a smooth Brownian terminal payoff
{c}
{title2=$\beta_t=\mathbb E[\phi'(W_T)\mid\mathcal F_t]\mathbf1_{\{t\le T\}}$}
In the completed natural <Brownian filtration>, a bounded continuously differentiable $\phi$ with bounded derivative satisfies
$$
\phi(W_T)=\mathbb E\phi(W_T)+\int_0^T\mathbb E[\phi'(W_T)\mid\mathcal F_t]\,dW_t.
$$
The integrand is unique up to $d\mathbb P\,dt$-almost everywhere equality. <Gaussian integration by parts> first gives the duality between the payoff and every square-integrable <Itô integral>. The <Brownian martingale representation theorem> then identifies the integrand as the projection of $\phi'(W_T)\mathbf1_{\{t\le T\}}$ onto the <predictable processes> in the product $L^2$ space. This explains the <conditional expectation> in the formula, rather than a nonadapted terminal derivative.
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