Malliavin calculus

Definition

Let $T > 0$, let $(B_t)_{0 \le t \le T}$ be a Brownian motion on $(\Omega, \mathcal{F}, P)$, and let $\mathcal{F}$ be generated by $B$. Let $H = L^2([0, T])$, and for $h \in H$ let $W(h) = \int_0^T h(t) \, dB_t$ be the Itô integral. A smooth random variable is $F = f(W(h_1), \ldots, W(h_n))$ with $h_1, \ldots, h_n \in H$ and $f \in C^\infty(\R^n)$, where $f$ and all its partial derivatives have polynomial growth. Its Malliavin derivative is the $H$-valued random variable $$D_t F = \sum_{i=1}^n \partial_i f(W(h_1), \ldots, W(h_n)) \, h_i(t), \quad 0 \le t \le T.$$ The operator $D$ is closable from $L^2(\Omega)$ to $L^2(\Omega; H)$. Its closure is again written $D$, and its domain $\mathbb{D}^{1,2}$ is the completion of the smooth random variables for the norm $\norm{F}_{1,2}^2 = E[F^2] + E\left[\norm{DF}_H^2\right]$. Malliavin calculus is the study of $D$, of its adjoint $\delta$, called the Skorokhod integral, and of their applications.