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

Yeliz Yolcu Okur
2010 Stochastic Analysis and Applications  
We proved white noise generalization of the Clark-Ocone formula under change of measure by using white noise analysis and Malliavin calculus. Let W (t) be a Brownian motion on the filtered white noise probability space (Ω, B, {Ft} t≥0 , P ) and letŴ (t) be defined as dŴ (t) = u(t) + dW (t), where u(t) is an Ft-measurable process satisfying certain conditions. Let Q be the probability measure equivalent P such thatŴ (t) is a Brownian motion with respect to Q, in virtue of the Girsanov theorem.
more » ... this paper, it is shown that for any random variable F ∈ L 2 (P ) where EQ is the expectation under Q and DtF (ω) is the (Hida) Malliavin derivative. The important point to note here is in this settlement F need not belong to stochastic Sobolev space,
doi:10.1080/07362994.2010.515498 fatcat:i4ysa57sb5e5hpxhun3avcxbsi