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In
mathematics , the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula ) is a
theorem of
stochastic analysis . It expresses the value of some
function F defined on the
classical Wiener space of continuous paths starting at the origin as the sum of its
mean value and an
Itô integral with respect to that path. It is named after the contributions of
mathematicians J.M.C. Clark (1970),
Daniel Ocone (1984) and U.G. Haussmann (1978).
Statement of the theorem
Let C 0 ([0, T ]; R ) (or simply C 0 for short) be classical Wiener space with Wiener measure γ . Let F : C 0 → R be a BC1 function, i.e. F is
bounded and
Fréchet differentiable with bounded derivative DF : C 0 → Lin(C 0 ; R ). Then
F
(
σ
)
=
∫
C
0
F
(
p
)
d
γ
(
p
)
+
∫
0
T
E
∂
∂
t
∇
H
F
(
−
)
|
Σ
t
(
σ
)
d
σ
t
.
{\displaystyle F(\sigma )=\int _{C_{0}}F(p)\,\mathrm {d} \gamma (p)+\int _{0}^{T}\mathbf {E} \left[\left.{\frac {\partial }{\partial t}}\nabla _{H}F(-)\right|\Sigma _{t}\right](\sigma )\,\mathrm {d} \sigma _{t}.}
In the above
F (σ ) is the value of the function F on some specific path of interest, σ ;
the first integral,
∫
C
0
F
(
p
)
d
γ
(
p
)
=
E
F
{\displaystyle \int _{C_{0}}F(p)\,\mathrm {d} \gamma (p)=\mathbf {E} [F]}
is the
expected value of F over the whole of Wiener space C 0 ;
∫
0
T
⋯
d
σ
(
t
)
{\displaystyle \int _{0}^{T}\cdots \,\mathrm {d} \sigma (t)}
is an
Itô integral ;
More generally, the conclusion holds for any F in L 2 (C 0 ; R ) that is differentiable in the sense of Malliavin.
Integration by parts on Wiener space
The Clark–Ocone theorem gives rise to an
integration by parts formula on classical Wiener space, and to write
Itô integrals as
divergences :
Let B be a standard Brownian motion, and let L 0 2,1 be the Cameron–Martin space for C 0 (see
abstract Wiener space . Let V : C 0 → L 0 2,1 be a
vector field such that
V
˙
=
∂
V
∂
t
:
0
,
T
×
C
0
→
R
{\displaystyle {\dot {V}}={\frac {\partial V}{\partial t}}:[0,T]\times C_{0}\to \mathbb {R} }
is in L 2 (B ) (i.e. is
Itô integrable , and hence is an
adapted process ). Let F : C 0 → R be BC1 as above. Then
∫
C
0
D
F
(
σ
)
(
V
(
σ
)
)
d
γ
(
σ
)
=
∫
C
0
F
(
σ
)
(
∫
0
T
V
˙
t
(
σ
)
d
σ
t
)
d
γ
(
σ
)
,
{\displaystyle \int _{C_{0}}\mathrm {D} F(\sigma )(V(\sigma ))\,\mathrm {d} \gamma (\sigma )=\int _{C_{0}}F(\sigma )\left(\int _{0}^{T}{\dot {V}}_{t}(\sigma )\,\mathrm {d} \sigma _{t}\right)\,\mathrm {d} \gamma (\sigma ),}
i.e.
∫
C
0
⟨
∇
H
F
(
σ
)
,
V
(
σ
)
⟩
L
0
2
,
1
d
γ
(
σ
)
=
−
∫
C
0
F
(
σ
)
div
(
V
)
(
σ
)
d
γ
(
σ
)
{\displaystyle \int _{C_{0}}\left\langle \nabla _{H}F(\sigma ),V(\sigma )\right\rangle _{L_{0}^{2,1}}\,\mathrm {d} \gamma (\sigma )=-\int _{C_{0}}F(\sigma )\operatorname {div} (V)(\sigma )\,\mathrm {d} \gamma (\sigma )}
or, writing the integrals over C 0 as expectations:
E
⟨
∇
H
F
,
V
⟩
=
−
E
F
div
V
,
{\displaystyle \mathbb {E} {\big [}\langle \nabla _{H}F,V\rangle {\big ]}=-\mathbb {E} {\big [}F\operatorname {div} V{\big ]},}
where the "divergence" div(V ) : C 0 → R is defined by
div
(
V
)
(
σ
)
:=
−
∫
0
T
V
˙
t
(
σ
)
d
σ
t
.
{\displaystyle \operatorname {div} (V)(\sigma ):=-\int _{0}^{T}{\dot {V}}_{t}(\sigma )\,\mathrm {d} \sigma _{t}.}
The interpretation of stochastic integrals as divergences leads to concepts such as the
Skorokhod integral and the tools of the
Malliavin calculus .
See also
References
External links