The Malliavin derivative of a time integral is a key concept in Malliavin calculus, particularly when dealing with random variables that depend on an integral over time, such as in stochastic processes or solutions to stochastic differential equations (SDEs).
Let’s start by defining a time integral and then compute its Malliavin derivative.
Suppose $W(t)$ is a standard Brownian motion on $[0, T]$ defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ , and we are interested in the Malliavin derivative of a random variable $Y$ given by an integral over time. For example, consider the following random variable:
$Y = \int_0^T f(t) \, dW(t),$
where $f(t)$ is some deterministic function, and $W(t)$ is the Brownian motion. The goal is to compute the Malliavin derivative of $Y$ , $D Y$ , which gives us the sensitivity of $Y$ to perturbations in the Brownian path.
The Malliavin derivative of $Y = \int_0^T f(t) \, dW(t)$ is computed by applying the definition of the Malliavin derivative. The Malliavin derivative $D_s Y$ is defined as the Fréchet derivative of $Y$ with respect to the Brownian motion at time $s$ .
By the definition of the Malliavin derivative, we have:
$D_s Y = D_s \left( \int_0^T f(t) \, dW(t) \right).$
Using the properties of the Malliavin derivative, particularly the linearity and the fact that the derivative with respect to a stochastic integral "picks out" the integrand at the point $s$ , we obtain:
$D_s Y = f(s),$
where $f(s)$ is the value of the function $f(t)$ evaluated at time $t = s$ .
If we have a more general time integral, say:
$Y = \int_0^T g(t) \, dW(t) + \int_0^T h(t) \, dB(t),$
where $W(t)$ and $B(t)$ are two independent Brownian motions, and $g(t)$ and $h(t)$ are deterministic functions, the Malliavin derivative with respect to $W(t)$ is given by:
$D_s Y = g(s),$
and the Malliavin derivative with respect to $B(t)$ is:
$D_s Y = h(s),$
where $D_s$ picks out the integrand associated with the respective Brownian motion.