Explicit formulae for time-space Brownian chaos

Let F be a square-integrable and infinitely weakly differentiable functional of a standard Brownian motion X: we show that the nth integrand in the time-space chaotic decomposition of F has the form E(alpha((n))D(n)F\X-t1,...,X-tn), where alpha((n)) is a transform of Hardy type and D-n denotes the nth 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.