Clark-Ocone formula by the S-transform on the Poisson white noise space

Abstract

Given ' a square-integrable Poisson white noise functionals we show that the Segal-Bargmann (S-) transform t 㜀! S'(Pt(g)) is absolutely continuous with d dt S '(P t(g)) = S @ � E(@t'jFt ) � (g) for almost all t 2 R, where Pt(g) = 1(1 , t) �g for g in the Schwartz space S on R, and @t means the Poisson white noise derivative. After integration with respect to t and applying the inverse S-transform, this identity recovers the Clark-Ocone formula for '.