3
$\begingroup$

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).

$\endgroup$
1

1 Answer 1

4
$\begingroup$

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.)

$\endgroup$
1
  • $\begingroup$ 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, 2013 at 2:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.