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.

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:

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

• H-derivative