Cameron–Martin theorem

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In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.

Motivation

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The standard Gaussian measure   on  -dimensional Euclidean space   is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the  -dimensional Lebesgue measure, denoted here  .) Instead, a measurable subset   has Gaussian measure

 

Here   refers to the standard Euclidean dot product in  . The Gaussian measure of the translation of   by a vector   is

 

So under translation through  , the Gaussian measure scales by the distribution function appearing in the last display:

 

The measure that associates to the set   the number   is the pushforward measure, denoted  . Here   refers to the translation map:  . The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by

 

The abstract Wiener measure   on a separable Banach space  , where   is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace  .

Statement of the theorem

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Let   be an abstract Wiener space with abstract Wiener measure  . For  , define   by  . Then   is equivalent to   with Radon–Nikodym derivative

 

where

 

denotes the Paley–Wiener integral.

The Cameron–Martin formula is valid only for translations by elements of the dense subspace  , called Cameron–Martin space, and not by arbitrary elements of  . If the Cameron–Martin formula did hold for arbitrary translations, it would contradict the following result:

If   is a separable Banach space and   is a locally finite Borel measure on   that is equivalent to its own push forward under any translation, then either   has finite dimension or   is the trivial (zero) measure. (See quasi-invariant measure.)

In fact,   is quasi-invariant under translation by an element   if and only if  . Vectors in   are sometimes known as Cameron–Martin directions.

Integration by parts

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The Cameron–Martin formula gives rise to an integration by parts formula on  : if   has bounded Fréchet derivative  , integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives

 

for any  . Formally differentiating with respect to   and evaluating at   gives the integration by parts formula

 

Comparison with the divergence theorem of vector calculus suggests

 

where   is the constant "vector field"   for all  . The wish to consider more general vector fields and to think of stochastic integrals as "divergences" leads to the study of stochastic processes and the Malliavin calculus, and, in particular, the Clark–Ocone theorem and its associated integration by parts formula.

An application

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Using Cameron–Martin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a   symmetric non-negative definite matrix   whose elements   are continuous and satisfy the condition

 

it holds for a  −dimensional Wiener process   that

 

where   is a   nonpositive definite matrix which is a unique solution of the matrix-valued Riccati differential equation

 

with the boundary condition  .

In the special case of a one-dimensional Brownian motion where  , the unique solution is  , and we have the original formula as established by Cameron and Martin:  

See also

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References

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  • Cameron, R. H.; Martin, W. T. (1944). "Transformations of Wiener Integrals under Translations". Annals of Mathematics. 45 (2): 386–396. doi:10.2307/1969276. JSTOR 1969276.
  • Liptser, R. S.; Shiryayev, A. N. (1977). Statistics of Random Processes I: General Theory. Springer-Verlag. ISBN 3-540-90226-0.