Malliavin calculus for Hilbert space-valued random variables

Malliavin calculus, often referred to as the stochastic calculus of variations, is a powerful tool for studying the smoothness properties of random variables and has applications in stochastic analysis, particularly in proving the existence of densities and in sensitivity analysis of stochastic processes. Extending Malliavin calculus to Hilbert space-valued random variables involves working within a framework that generalizes classical results to infinite-dimensional spaces.

1. Hilbert Space-Valued Random Variables

A Hilbert space-valued random variable $X$ is a mapping from a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to a separable Hilbert space $H$ such that $X(\omega) \in H$ for all $\omega \in \Omega$ . Typically, $H$ could be $L^2([0, T])$ , a space of square-integrable functions, or more complex spaces encountered in applications like solutions to PDEs or SPDEs.

2. Extension of Malliavin Calculus

Malliavin calculus for Hilbert space-valued random variables extends the classical Malliavin calculus (developed initially for real-valued random variables) to handle these infinite-dimensional settings. This extension allows the analysis of the regularity properties of random processes taking values in Hilbert spaces, such as stochastic processes governed by SPDEs.

Key Concepts in Malliavin Calculus for Hilbert Spaces

Malliavin Derivative: If $X$ is a Hilbert space-valued random variable, its Malliavin derivative $D X$ is a mapping that takes $X$ to a Hilbert space $H$ -valued process:

$D X: \Omega \rightarrow L^2([0, T]; H).$

The derivative $D X(\omega)(t)$ represents the infinitesimal change in $X$ due to a perturbation in the underlying Brownian path at time $t$ .

Sobolev Spaces of Random Variables: The space $\mathbb{D}^{1,2}(H)$ consists of $H$ -valued random variables $X$ such that:

$\mathbb{E}[\|X\|H^2] < \infty \quad \text{and} \quad \mathbb{E}[\|D X\|{L^2([0, T]; H)}^2] < \infty.$

Skorohod Integral: The adjoint operator to the Malliavin derivative, which acts similarly to an Itô integral but allows for integration with respect to more general processes. For a Hilbert space-valued process $u$ , the Skorohod integral $\delta(u)$ is defined when $u$ is sufficiently regular and belongs to the domain $\text{Dom}(\delta)$ .

3. Framework and Operators

Malliavin Derivative Operator $D$ : For an $H$ -valued random variable $X$ that is Malliavin differentiable, $D X$ maps into $L^2(\Omega \times [0, T]; H)$ . This operator extends to derivatives of higher orders $D^k$ when analyzing the smoothness and differentiability of $X$ in a more refined way.
Divergence Operator $\delta$ : The divergence operator $\delta$ acts as an extension of the Itô integral to processes that are not necessarily adapted. For an $H$ -valued process $u$ , $\delta(u)$ is defined when $u$ satisfies certain integrability and adaptiveness properties.

4. Applications

Stochastic PDEs (SPDEs): Malliavin calculus is particularly valuable in studying SPDEs where solutions are Hilbert space-valued processes. It provides techniques for analyzing the regularity of solutions and proving the existence of smooth densities for their probability distributions.
Sensitivity Analysis: Malliavin calculus allows for the computation of sensitivities (Greeks) in mathematical finance when modeling assets as processes in infinite-dimensional spaces.
Probability Densities: The calculus is used to establish that certain Hilbert space-valued random variables have smooth probability density functions with respect to Lebesgue measure. This is particularly important in proving the non-degeneracy of solutions to stochastic equations.

5. Technical Conditions and Results

Clark-Ocone Formula: For Hilbert space-valued random variables, an extension of the Clark-Ocone formula expresses a random variable $X$ in terms of its Malliavin derivative and the Skorohod integral:

$X = \mathbb{E}[X] + \delta(D X).$