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Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$. Let $f(x)$ be $C^1$ function such that $|f'(x)| > 0$. Then if $\xi$ is a random variable measurable w.r.t. $\mathcal{F}_T$, process $M_t=E(f(\xi)|\mathcal{F_t})$ is a martingale and according to martingale representation theorem there exists an adapted process $a(t)$ such that $$ M_t = E(f(\xi)) + \int_0^ta(s)dW_s. $$

My question is if it is possible to express $a(t)$ as $a(t) = A(t,W_t)$ or $a(t) = A(t,M_t)$ where $A(t,x)$ is deterministic (maybe in some special cases).

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No, but under some regularity conditions you might represent $a(t)$ in terms of Malliavin calculus by means of the Clark-Ocone formula (see e.g. the Lecture notes by Eulalia Nualart, Section 1.5.3.)

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  • $\begingroup$ Thanks, that helped. I wasn't familiar with Malliavin calculus, so it took some time for me to figure out proper conditions on $\xi$ and $f$ $\endgroup$ – niyazets Feb 19 '13 at 2:57

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