Explicit formulae for time-space Brownian chaos

Abstract

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.