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

Abstract

Abstract. 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. In 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, D1,2 which is subset of L 2 (P ). This makes this formula more useful in applications to finance. For example, the replicating portfolio for a digital option, whose payoff function χ [K,∞) W (T ) / ∈ D1,2, is calculated by using this generalized Clark-Ocone formula under change of measure.

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