The Malliavin derivative is a powerful tool in stochastic calculus that allows us to analyze the smoothness of probability densities of random variables that are functionals of stochastic processes, particularly Wiener processes.
Malliavin calculus provides a framework for analyzing the smoothness of probability densities of functionals of stochastic processes. The goal is to determine conditions under which a random variable $F$ has a smooth density function $p_F(x)$ with respect to the Lebesgue measure.
Let $\left(W_t\right)_{t \geq 0}$ be a standard Wiener process on a probability space $(\Omega, F , P)$. For a functional $F=f(W)$, the Malliavin derivative $D_t F$ measures the sensitivity of $F$ to small perturbations of the Brownian motion at time $t$.
Formally, the Malliavin derivative is defined as:
$$ D_t F=\lim {\epsilon \rightarrow 0} \frac{F\left(W+\epsilon 1{[t, T]}\right)-F(W)}{\epsilon} $$
if the limit exists in $L^2(\Omega)$.
For $F=W_T$, the Malliavin derivative is simply:
$$ D_t W_T= 1 _{[0, T]}(t) $$
since a perturbation in $W_t$ propagates to $W_T$ for $t \leq T$.
Consider the stochastic differential equation:
$$ d X_t=b\left(X_t\right) d t+\sigma\left(X_t\right) d W_t, \quad X_0=x_0 $$
The solution at time $T$ is denoted by $F=X_T$, and we are interested in the smoothness of the density $p_F(x)$.
To apply Malliavin calculus, we compute the Malliavin derivative:
$$ D_s X_T=\sigma\left(X_s\right)+\int_s^T \frac{\partial b}{\partial x}\left(X_u\right) D_s X_u d u+\int_s^T \frac{\partial \sigma}{\partial x}\left(X_u\right) D_s X_u d W_u $$
Under appropriate non-degeneracy conditions on $\sigma\left(X_t\right)$, this derivative allows us to study the density $p_F(x)$.
The main result comes from the Malliavin regularity criterion, which states that if: