MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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).

share|cite|improve this question
up vote 2 down vote accepted

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.)

share|cite|improve this answer
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 $f$ – niyazets Feb 19 '13 at 2:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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