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In mathematics , the Malliavin derivative 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
H
{\displaystyle H}
be the Cameron–Martin space , and
C
0
{\displaystyle C_{0}}
denote classical Wiener space :
H
:=
{
f
∈
W
1
,
2
(
[
0
,
T
]
;
R
n
)
|
f
(
0
)
=
0
}
:=
{
paths starting at 0 with first derivative in
L
2
}
{\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}\}}
;
C
0
:=
C
0
(
[
0
,
T
]
;
R
n
)
:=
{
continuous paths starting at 0
}
;
{\displaystyle C_{0}:=C_{0}([0,T];\mathbb {R} ^{n}):=\{{\text{continuous paths starting at 0}}\};}
By the Sobolev embedding theorem ,
H
⊂
C
0
{\displaystyle H\subset C_{0}}
. Let
i
:
H
→
C
0
{\displaystyle i:H\to C_{0}}
denote the inclusion map .
Suppose that
F
:
C
0
→
R
{\displaystyle F:C_{0}\to \mathbb {R} }
is Fréchet differentiable . Then the Fréchet derivative is a map
D
F
:
C
0
→
L
i
n
(
C
0
;
R
)
;
{\displaystyle \mathrm {D} F:C_{0}\to \mathrm {Lin} (C_{0};\mathbb {R} );}
i.e., for paths
σ
∈
C
0
{\displaystyle \sigma \in C_{0}}
,
D
F
(
σ
)
{\displaystyle \mathrm {D} F(\sigma )\;}
is an element of
C
0
∗
{\displaystyle C_{0}^{*}}
, the dual space to
C
0
{\displaystyle C_{0}\;}
. Denote by
D
H
F
(
σ
)
{\displaystyle \mathrm {D} _{H}F(\sigma )\;}
the continuous linear map
H
→
R
{\displaystyle H\to \mathbb {R} }
defined by
D
H
F
(
σ
)
:=
D
F
(
σ
)
∘
i
:
H
→
R
,
{\displaystyle \mathrm {D} _{H}F(\sigma ):=\mathrm {D} F(\sigma )\circ i:H\to \mathbb {R} ,}
sometimes known as the H -derivative . Now define
∇
H
F
:
C
0
→
H
{\displaystyle \nabla _{H}F:C_{0}\to H}
to be the adjoint of
D
H
F
{\displaystyle \mathrm {D} _{H}F\;}
in the sense that
∫
0
T
(
∂
t
∇
H
F
(
σ
)
)
⋅
∂
t
h
:=
⟨
∇
H
F
(
σ
)
,
h
⟩
H
=
(
D
H
F
)
(
σ
)
(
h
)
=
lim
t
→
0
F
(
σ
+
t
i
(
h
)
)
−
F
(
σ
)
t
.
{\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
D
t
{\displaystyle \mathrm {D} _{t}}
is defined by
(
D
t
F
)
(
σ
)
:=
∂
∂
t
(
(
∇
H
F
)
(
σ
)
)
.
{\displaystyle \left(\mathrm {D} _{t}F\right)(\sigma ):={\frac {\partial }{\partial t}}\left(\left(\nabla _{H}F\right)(\sigma )\right).}
The domain of
D
t
{\displaystyle \mathrm {D} _{t}}
is the set
F
{\displaystyle \mathbf {F} }
of all Fréchet differentiable real-valued functions on
C
0
{\displaystyle C_{0}\;}
; the codomain is
L
2
(
[
0
,
T
]
;
R
n
)
{\displaystyle L^{2}([0,T];\mathbb {R} ^{n})}
.
The Skorokhod integral
δ
{\displaystyle \delta \;}
is defined to be the adjoint of the Malliavin derivative:
δ
:=
(
D
t
)
∗
:
image
(
D
t
)
⊆
L
2
(
[
0
,
T
]
;
R
n
)
→
F
∗
=
L
i
n
(
F
;
R
)
.
{\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 [ ]
References [ ]