Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness properties for claim.) Now, I am interested to know the strategy (the integrand of stochastic integral) explicitly for given claim. Consider simple claim such as $f(B_T)$ where $f$ is smooth and bounded function. Is there any explicit formula which can give process $Z$ such that $\int_0^T Z_t dB_t = f(B_T)-\mathbb{E}[f(B_T)]$ ?
Attempt : I tried to use Ito formula to calculate $Z$. However it works only when $f$ is linear. In other cases, I don't have much progress. Although, Martingales Representation Theorem asserts the existence of $Z$, it doesn't provide a way to explicitly calculate it. I would like to know how to make calculations. Thanks!
This is a consequence from the Clarke-Ocone Theorem, and uses Malliavin derivative. See also the Clarke-Ocone formula paragraphe here. If you want a technical reference, see this introductory course, mainly theorem 1 p. 18.