---
tag: FQMAKKE8R9
kind: definition
name: Malliavin calculus
msc: [60H07]
defines: 1EC7VPHFAY
uses: [22735FGJV6, JA95NH206E, 5Z00K7K1HY, 2E8MQXDYNV]
sources:
  - url: https://en.wikipedia.org/wiki/Malliavin_calculus
---

## Definition

Let $T > 0$, let $(B_t)_{0 \le t \le T}$ be a [[Brownian motion|22735FGJV6]] 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|JA95NH206E]].
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.

## Remarks

Example: $F = B_T^2 = W(\mathbf{1}_{[0,T]})^2$ is smooth with $f(x) = x^2$, so $D_t F = 2 B_T$ for $0 \le t \le T$.
Non-example: $F = \mathbf{1}_{\set{B_T > 0}}$ is not in $\mathbb{D}^{1,2}$; an indicator of an event $A$ lies in $\mathbb{D}^{1,2}$ only if $P(A)$ is $0$ or $1$.
The basic integration by parts formula is $E[\langle DF, h \rangle_H] = E[F \, W(h)]$ for smooth $F$ and $h \in H$.
The Clark–Ocone formula $F = E[F] + \int_0^T E[D_t F \mid \mathcal{F}_t] \, dB_t$ for $F \in \mathbb{D}^{1,2}$ makes the [[martingale representation theorem|2E8MQXDYNV]] explicit.
The setting is often the [[classical Wiener space|5Z00K7K1HY]] of continuous paths.