Malliavin Calculus - Clark-Ocone Formula

Clark-Ocone Formula

One of the most useful results from Malliavin calculus is the Clark-Ocone theorem, which allows the process in the martingale representation theorem to be identified explicitly. A simplified version of this theorem is as follows:

For satisfying which is Lipschitz and such that F has a strong derivative kernel, in the sense that for in C

then

where H is the previsible projection of F'(x, (t,1]) which may be viewed as the derivative of the function F with respect to a suitable parallel shift of the process X over the portion (t,1] of its domain.

This may be more concisely expressed by

Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals F by replacing the derivative kernel used above by the "Malliavin derivative" denoted in the above statement of the result.