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

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Okur Y. Y.

STOCHASTIC ANALYSIS AND APPLICATIONS, vol.28, no.6, pp.1106-1121, 2010 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 28 Issue: 6
  • Publication Date: 2010
  • Doi Number: 10.1080/07362994.2010.515498
  • Journal Name: STOCHASTIC ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1106-1121
  • Keywords: Change of measure, Clark-Ocone formula, (Hida) Malliavin derivative, Malliavin calculus
  • Middle East Technical University Affiliated: Yes

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 (Omega, B, {F(t)}(0 <= t <= T), P) and let (W) over cap (t) be defined as d (W) over cap (t) = u(t)dt + dW (t), where u W(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 (W) over cap 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,