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). 



On martingale representation theorem
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](http://www.math.wisc.edu/~kurtz/NualartLectureNotes.pdf) by Eulalia Nualart, Section 1.5.3.)


