## Bernoulli

- Bernoulli
- Volume 9, Number 1 (2003), 25-48.

### 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.

#### Article information

**Source**

Bernoulli, Volume 9, Number 1 (2003), 25-48.

**Dates**

First available in Project Euclid: 6 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1068129009

**Digital Object Identifier**

doi:10.3150/bj/1068129009

**Mathematical Reviews number (MathSciNet)**

MR1963671

**Zentralblatt MATH identifier**

1033.60087

**Keywords**

Brownian bridge Brownian motion Clark-Ocone formula enlargement of filtrations Hardy operators static hedging Stroock's formula time-space chaos

#### Citation

Peccati, Giovanni. Explicit formulae for time-space Brownian chaos. Bernoulli 9 (2003), no. 1, 25--48. doi:10.3150/bj/1068129009. https://projecteuclid.org/euclid.bj/1068129009