An Extension of the Clark-Ocone Formula Under Benchmark Measure for Levy Processes

Abstract

The classical Clark-Ocone theorem states that any random variable F is an element of D-1,2(W) subset of L-2 (F-T, P) can be represented as

F = E[F] + integral(T)(0) E[DtF vertical bar F-t]dWd(t),

where E[.vertical bar F-t] denotes the conditional expectation, W(.) is a Brownian motion with canonical filtration {Ft}(t is an element of[0,T]) and D denotes the Malliavin derivative in the direction of W. Since many applications in financial mathematics require representation of random variables with respect to risk neutral martingale measure, an equivalent martingale measure version of this theorem was stated by Karatzas and Ocone (Stoch. Stoch. Rep. 34 (1991), 187-220). In this paper, we extend these results to be valid for square integrable pure jump Levy processes with no drift and for square integrable Ito-Levy processes using Malliavin calculus and white noise analysis. This extension might be useful for some applications in finance. As an application of our result, we calculate explicitly the closest hedge strategy for the digital option whose pay-off, F = chi([H,K))(S(T)) is not an element of D-1,2(W,(N) over tilde), is square integrable and the stock price S(.) is driven by a Levy process.