The chain rule for the Malliavin derivative provides a way to compute the derivative of a function of a random variable that is itself Malliavin differentiable. This rule is essential for analyzing the derivative of compositions of random variables in stochastic analysis.
Let $X$ be a real-valued or Hilbert space-valued random variable that is Malliavin differentiable, and let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ (or $\phi: H \rightarrow H'$ for some Hilbert spaces $H$ and $H'$ ) be a sufficiently smooth function. The goal is to compute the Malliavin derivative of $\phi(X)$ .
If $X = (X_1, X_2, \ldots, X_n)$ is an $n$ -dimensional vector of Malliavin differentiable random variables and $\phi$ is a $C^1$ function (continuously differentiable), then the chain rule for the Malliavin derivative is given by:
$D \phi(X) = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i}(X) \cdot D X_i,$
where $D X_i$ is the Malliavin derivative of $X_i$ . This rule extends to infinite-dimensional spaces with appropriate modifications involving Fréchet derivatives.
For a Hilbert space-valued random variable $X \in H$ and a function $\phi: H \rightarrow H'$ where $H$ and $H'$ are separable Hilbert spaces, the chain rule takes the form:
$D \phi(X) = D\phi(X) \circ D X,$
where $D\phi(X)$ is the Fréchet derivative of $\phi$ at $X$ and $D X$ is the Malliavin derivative of $X$ . This involves interpreting the composition as an application of linear operators in the space of $H$ -valued functions.
Consider $X(t) = W(t)$ , a Brownian motion, and $\phi(x) = x^2$ . The Malliavin derivative $D W(t)$ is given by: $D_s W(t) = \mathbf{1}_{[0, t]}(s),$
which indicates the sensitivity of $W(t)$ to perturbations at time $s$ . Applying the chain rule:
$D (\phi(W(t))) = 2W(t) \cdot D W(t) = 2W(t) \cdot \mathbf{1}_{[0, t]}(s).$
This expresses how the function $W(t)^2$ changes with respect to changes in the underlying Brownian motion.