I'm trying to understand a proof that requires Malliavan Calculus, but have no experience with the topic. My question revolves around showing that the Malliavan derivative of a geometric brownian motion is equal its volatility, i.e.
$$ dX_t = \mu X_t dt + \sigma X_t dW_t \\ \Rightarrow \mathcal{D_t}(X_t) = \sigma X_t $$
My understanding is that the Malliavan derivative is given by
$$ \mathcal{D_t}(F) = \sum_{n=0}^\infty n I_{n-1} (f_n) $$
where $I_n$ is the $n$-fold iterated ito integral and $f_n$ is the Wiener-Ito chaos expansion of $F$. I also understand that this expansion can be found by using a Hermite expansion, but this is where I'm stuck.
How does one carry out the expansion in a practical sense?
This answer comes from the lecture notes by David Nualart, contained in ch. 1 of "Stochastic Equations for Complex Systems: Theoretical and Computational Topics", Springer, 2015.
The Malliavan derivative is defined in the following way. Let $W$ be defined as the canonical process on the probability space $(\Omega, \mathcal{F}, P)$ where $\Omega = C([0, T])$ for some fixed $T$, $\mathcal{F}$ the Borel $\sigma$-field of $\Omega$, and $P$ the Wiener measure.
Define $W(h) = \int_0^T h(t) dW(t)$ for square integrable functions $h \in H = L^2([0, T])$. If $F$ is a smooth and cylindrical random variable of the form $$ F = f(W(h_1), ..., W(h_n)) $$ where $f \in C_p^\infty(\mathbb{R}^n)$, then the derivative $DF$ is the $H$-valued random variable defined by $$ D_t F = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(W(h_1), ..., W(h_n))h_i(t) $$
Now take the geometric brownian motion defined by $$ X_t = X_0 exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) $$ Here $h(s) = \textbf{1}_{[0, t]}$ and $\frac{\partial f}{\partial x} = X_t \sigma$. Thus the Malliavan derivative of the geometric brownian motion is $$ D_t X_t = \sigma X_t $$
Intuitively, this derivative is a calculus of variations with respect to trajectories of the Brownian motion, so you can think of how $X_t$ changes given a change in $W_s$, ie $D_s X_t$ can be calculated for any $s$ and any $t$.