Clark-Ocone theorem
Encyclopedia
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 Itō integral with respect to that path. It is named after the contributions of mathematician
s J.M.C. Clark (1970), Daniel Ocone
(1984) and U.G. Haussmann (1978).
and Fréchet differentiable
with bounded derivative DF : C0 → Lin(C0; R). Then
In the above
More generally, the conclusion holds for any F in L2(C0; R) that is differentiable in the sense of Malliavin.
formula on classical Wiener space, and to write Itō integrals as divergence
s:
Let B be a standard Brownian motion, and let L02,1 be the Cameron–Martin space for C0 (see abstract Wiener space
. Let V : C0 → L02,1 be a vector field
such that
is in L2(B) (i.e. is Itō integrable, and hence is an adapted process
). Let F : C0 → R be BC1 as above. Then
i.e.
or, writing the integrals over C0 as expectations:
where the "divergence" div(V) : C0 → R is defined by
The interpretation of stochastic integrals as divergences leads to concepts such as the Skorokhod integral
and the tools of the Malliavin calculus
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Clark–Ocone theorem (also known as the Clark–Ocone–Haussmann theorem or formula) is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
of stochastic analysis
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
. It expresses the value of some function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
F defined on the classical Wiener space
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
of continuous paths starting at the origin as the sum of its mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
value and an Itō integral with respect to that path. It is named after the contributions of mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s J.M.C. Clark (1970), Daniel Ocone
Daniel Ocone
Daniel L. Ocone is a Professor in the Mathematics Department at Rutgers University, where he specializes in probability theory and stochastic processes. He obtained his Ph.D at MIT in 1980 under the supervision of Sanjoy K. Mitter. He is known for the Clark–Ocone theorem...
(1984) and U.G. Haussmann (1978).
Statement of the theorem
Let C0([0, T]; R) (or simply C0 for short) be classical Wiener space with Wiener measure γ. Let F : C0 → R be a BC1 function, i.e. F is boundedBounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
and Fréchet differentiable
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
with bounded derivative DF : C0 → Lin(C0; R). Then
In the above
- F(σ) is the value of the function F on some specific path of interest, σ;
- the first integral,
-
- is the expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of F over the whole of Wiener space C0;
- the second integral,
-
- is an Itō integral;
- Σ∗ is the natural filtrationFiltrationFiltration is commonly the mechanical or physical operation which is used for the separation of solids from fluids by interposing a medium through which only the fluid can pass...
of Brownian motionBrownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
B : [0, T] × Ω → R: Σt is the smallest σ-algebra containing all Bs−1(A) for times 0 ≤ s ≤ t and Borel sets A ⊆ R;
- E[·|Σt] denotes conditional expectationConditional expectationIn probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....
with respect to the sigma algebra Σt;
- ∂/∂t denotes differentiationDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
with respect to time t; ∇H denotes the H-gradient; hence, ∂/∂t∇H is the Malliavin derivativeMalliavin derivativeIn 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...
.
More generally, the conclusion holds for any F in L2(C0; 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 partsIntegration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
formula on classical Wiener space, and to write Itō integrals as divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
s:
Let B be a standard Brownian motion, and let L02,1 be the Cameron–Martin space for C0 (see abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
. Let V : C0 → L02,1 be a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
such that
is in L2(B) (i.e. is Itō integrable, and hence is an adapted process
Adapted process
In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...
). Let F : C0 → R be BC1 as above. Then
i.e.
or, writing the integrals over C0 as expectations:
where the "divergence" div(V) : C0 → R is defined by
The interpretation of stochastic integrals as divergences leads to concepts such as the Skorokhod integral
Skorokhod integral
In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod...
and the tools of the Malliavin calculus
Malliavin calculus
The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....
.
See also
- Integral representation theorem for classical Wiener spaceIntegral representation theorem for classical Wiener spaceIn mathematics, the integral representation theorem for classical Wiener space is a result in the fields of measure theory and stochastic analysis...
, which uses the Clark–Ocone theorem in its proof - Integration by parts operatorIntegration by parts operatorIn mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications.-Definition:Let E...
- Malliavin calculusMalliavin calculusThe Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....