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February 2003 Explicit formulae for time-space Brownian chaos
Giovanni Peccati
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Bernoulli 9(1): 25-48 (February 2003). DOI: 10.3150/bj/1068129009


Let $F$ be a square-integrable and infinitely weakly differentiable functional of a standard Brownian motion $X$: we show that the $n$th integrand in the time-space chaotic decomposition of $F$ has the form $\mathbb{E}\lambdaeft(\alphapha _{\lambdaeft( n\right)}D^{n}F\mid X_{t_{1}},\rm dots,X_{t_{n}}\right)$, where $\alphapha _{\lambdaeft( n\right)}$ is a transform of Hardy type and $D^{n}$ denotes the $n$th derivative operator. In this way, we complete the results of previous papers, and provide a time-space counterpart to the classic Stroock formulae for Wiener chaos. Our main tool is an extension of the Clark--Ocone formula in the context of initially enlarged filtrations. We discuss an application to the static hedging of path-dependent options in a continuous-time financial model driven by $X$. A formal connection between our results and the orthogonal decomposition of the space of square-integrable functionals of a standard Brownian bridge -- as proved by Gosselin and Wurzbacher -- is also established.


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Giovanni Peccati. "Explicit formulae for time-space Brownian chaos." Bernoulli 9 (1) 25 - 48, February 2003.


Published: February 2003
First available in Project Euclid: 6 November 2003

zbMATH: 1033.60087
MathSciNet: MR1963671
Digital Object Identifier: 10.3150/bj/1068129009

Keywords: Brownian bridge , Brownian motion , Clark-Ocone formula , Enlargement of filtrations , Hardy operators , static hedging , Stroock's formula , time-space chaos

Rights: Copyright © 2003 Bernoulli Society for Mathematical Statistics and Probability

Vol.9 • No. 1 • February 2003
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