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

Abstract

We prove the white noise generalization of the Clark-Ocone formula under change of measure by using Gaussian white noise analysis and Malliavin calculus. Let W(t) be a Brownian motion on the filtered white noise probability space (Ω, ℬ, {ℱ _t }_0≤t≤T , P) and let be defined as , where u(t) is an ℱ _t -measurable process satisfying certain conditions for all 0 ≤ t ≤ T. Let Q be the probability measure equivalent to P such that is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. In this article, it is shown that for any square integrable ℱ _T -measurable random variable,

where 𝔼 _Q is the expectation under Q and D _· F(ω) is the (Hida) Malliavin derivative. The important point in this settlement is F does not have to be in stochastic Sobolev space 𝔻_{1, 2} ⊂ L ²(P). This makes the formula more useful in applications of finance. As an example, the replicating portfolio for a digital option with the payoff χ_{[K, ∞)} W(T) ∉ 𝔻_{1, 2} is calculated by using this generalized Clark-Ocone formula under change of measure.

Keywords:

• Change of measure
• Clark-Ocone formula
• (Hida) Malliavin derivative
• Malliavin calculus

Mathematics Subject Classification:

• Primary: 60H07, 60H40
• Secondary: 60G20

The author wishes to express her thanks to Prof. Bernt Øksendal for suggestion of the problem and all the valuable comments.