Malliavin derivative

In mathematics, the Malliavin derivative^[1] is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense. ^{[citation needed]}

Definition

Let ${\displaystyle H}$ be the Cameron–Martin space, and ${\displaystyle C_{0}}$ denote classical Wiener space:

${\displaystyle H:=\{f\in W^{1,2}([0,T];\mathbb {R} ^{n})\;|\;f(0)=0\}:=\{{\text{paths starting at 0 with first derivative in }}L^{2}\}}$;

${\displaystyle C_{0}:=C_{0}([0,T];\mathbb {R} ^{n}):=\{{\text{continuous paths starting at 0}}\};}$

By the Sobolev embedding theorem, ${\displaystyle H\subset C_{0}}$. Let

${\displaystyle i:H\to C_{0}}$

denote the inclusion map.

Suppose that ${\displaystyle F:C_{0}\to \mathbb {R} }$ is Fréchet differentiable. Then the Fréchet derivative is a map

${\displaystyle \mathrm {D} F:C_{0}\to \mathrm {Lin} (C_{0};\mathbb {R} );}$

i.e., for paths ${\displaystyle \sigma \in C_{0}}$, ${\displaystyle \mathrm {D} F(\sigma )\;}$ is an element of ${\displaystyle C_{0}^{*}}$, the dual space to ${\displaystyle C_{0}\;}$. Denote by ${\displaystyle \mathrm {D} _{H}F(\sigma )\;}$ the continuous linear map ${\displaystyle H\to \mathbb {R} }$ defined by

${\displaystyle \mathrm {D} _{H}F(\sigma ):=\mathrm {D} F(\sigma )\circ i:H\to \mathbb {R} ,}$

sometimes known as the H-derivative. Now define ${\displaystyle \nabla _{H}F:C_{0}\to H}$ to be the adjoint of ${\displaystyle \mathrm {D} _{H}F\;}$ in the sense that

${\displaystyle \int _{0}^{T}\left(\partial _{t}\nabla _{H}F(\sigma )\right)\cdot \partial _{t}h:=\langle \nabla _{H}F(\sigma ),h\rangle _{H}=\left(\mathrm {D} _{H}F\right)(\sigma )(h)=\lim _{t\to 0}{\frac {F(\sigma +ti(h))-F(\sigma )}{t}}.}$

Then the Malliavin derivative ${\displaystyle \mathrm {D} _{t}}$ is defined by

${\displaystyle \left(\mathrm {D} _{t}F\right)(\sigma ):={\frac {\partial }{\partial t}}\left(\left(\nabla _{H}F\right)(\sigma )\right).}$

The domain of ${\displaystyle \mathrm {D} _{t}}$ is the set ${\displaystyle \mathbf {F} }$ of all Fréchet differentiable real-valued functions on ${\displaystyle C_{0}\;}$; the codomain is ${\displaystyle L^{2}([0,T];\mathbb {R} ^{n})}$.

The Skorokhod integral ${\displaystyle \delta \;}$ is defined to be the adjoint of the Malliavin derivative:

${\displaystyle \delta$ :=\left(\mathrm {D} _{t}\right)^{*}:\operatorname {image} \left(\mathrm {D} _{t}\right)\subseteq L^{2}([0,T];\mathbb {R} ^{n})\to \mathbf {F} ^{*}=\mathrm {Lin} (\mathbf {F} ;\mathbb {R} ).}

See also

• H-derivative