White Noise Generalization of the Clark-Ocone Formula Under Change of Measure

Yeliz Yolcu Okur
2010 Stochastic Analysis and Applications  
We prove white noise generalization of the Clark-Ocone formula under change of measure by using white noise analysis on Malliavin calculus. In this paper, it is shown that for any random variable F ∈ L 2 (P ) where E Q is the expectation under white noise measure Q,Ŵ (t) is the 1-dimensional Brownian motion constructed on the white noise probability space (Ω, B, Q) and D t F (ω) is the (Hida) Malliavin derivative. The important point to note here is in this settlement F should not belong to
more » ... d not belong to stochastic Sobolev space, D 1,2 which is subset of L 2 (P ) that leads this formula more useful in applications to finance. Moreover, the replicating portfolio for digital option, whose payoff function χ [K,∞) W (T ) / ∈ D 1,2 , is calculated by using generalized Clark-Ocone formula under change of measure. Here E[F ] denotes the generalized expectation, D t F (ω) = dF dω is the (generalized) Malliavin derivative, is the Wick product andẆ (t) is 1-dimensional Gaussian white noise. This formula holds for all F ∈ G * ⊃ L 2 (P ), where G * is a space of stochastic distributions and P is the white noise probability measure.
doi:10.1080/07362994.2010.515498 fatcat:i4ysa57sb5e5hpxhun3avcxbsi