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 separableBanach 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 densesubspace.
Statement of the theorem
Let be an abstract Wiener space with abstract Wiener measure . For , define by . Then is equivalent to with Radon–Nikodym derivative
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
The Cameron–Martin formula gives rise to an integration by parts formula on : if has boundedFréchet 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
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
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
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. ISBN3-540-90226-0.
February 02, 2023
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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 Contents 1 Motivation 2 Statement of the theorem 3 Integration by parts 4 An application 5 See also 6 ReferencesMotivation EditThe standard Gaussian measure g n displaystyle gamma n on n displaystyle n dimensional Euclidean space R n displaystyle mathbf R n is not translation invariant In fact there is a unique translation invariant Radon measure up to scale by Haar s theorem the n displaystyle n dimensional Lebesgue measure denoted here d x displaystyle dx Instead a measurable subset A displaystyle A has Gaussian measure g n A 1 2 p n 2 A exp 1 2 x x R n d x displaystyle gamma n A frac 1 2 pi n 2 int A exp left tfrac 1 2 langle x x rangle mathbf R n right dx Here x x R n displaystyle langle x x rangle mathbf R n refers to the standard Euclidean dot product in R n displaystyle mathbf R n The Gaussian measure of the translation of A displaystyle A by a vector h R n displaystyle h in mathbf R n is g n A h 1 2 p n 2 A exp 1 2 x h x h R n d x 1 2 p n 2 A exp 2 x h R n h h R n 2 exp 1 2 x x R n d x displaystyle begin aligned gamma n A h amp frac 1 2 pi n 2 int A exp left tfrac 1 2 langle x h x h rangle mathbf R n right dx 4pt amp frac 1 2 pi n 2 int A exp left frac 2 langle x h rangle mathbf R n langle h h rangle mathbf R n 2 right exp left tfrac 1 2 langle x x rangle mathbf R n right dx end aligned So under translation through h displaystyle h the Gaussian measure scales by the distribution function appearing in the last display exp 2 x h R n h h R n 2 exp x h R n 1 2 h R n 2 displaystyle exp left frac 2 langle x h rangle mathbf R n langle h h rangle mathbf R n 2 right exp left langle x h rangle mathbf R n tfrac 1 2 h mathbf R n 2 right The measure that associates to the set A displaystyle A the number g n A h displaystyle gamma n A h is the pushforward measure denoted T h g n displaystyle T h gamma n Here T h R n R n displaystyle T h mathbf R n to mathbf R n refers to the translation map T h x x h displaystyle T h x x h The above calculation shows that the Radon Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by d T h g n d g n x exp h x R n 1 2 h R n 2 displaystyle frac mathrm d T h gamma n mathrm d gamma n x exp left left langle h x right rangle mathbf R n tfrac 1 2 h mathbf R n 2 right The abstract Wiener measure g displaystyle gamma on a separable Banach space E displaystyle E where i H E displaystyle i H to E 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 i H E displaystyle i H subseteq E Statement of the theorem EditLet i H E displaystyle i H to E be an abstract Wiener space with abstract Wiener measure g Borel E 0 1 displaystyle gamma operatorname Borel E to 0 1 For h H displaystyle h in H define T h E E displaystyle T h E to E by T h x x i h displaystyle T h x x i h Then T h g displaystyle T h gamma is equivalent to g displaystyle gamma with Radon Nikodym derivative d T h g d g x exp h x 1 2 h H 2 displaystyle frac mathrm d T h gamma mathrm d gamma x exp left langle h x rangle sim tfrac 1 2 h H 2 right where h x i h x displaystyle langle h x rangle sim i h x denotes the Paley Wiener integral The Cameron Martin formula is valid only for translations by elements of the dense subspace i H E displaystyle i H subseteq E called Cameron Martin space and not by arbitrary elements of E displaystyle E If the Cameron Martin formula did hold for arbitrary translations it would contradict the following result If E displaystyle E is a separable Banach space and m displaystyle mu is a locally finite Borel measure on E displaystyle E that is equivalent to its own push forward under any translation then either E displaystyle E has finite dimension or m displaystyle mu is the trivial zero measure See quasi invariant measure In fact g displaystyle gamma is quasi invariant under translation by an element v displaystyle v if and only if v i H displaystyle v in i H Vectors in i H displaystyle i H are sometimes known as Cameron Martin directions Integration by parts EditThe Cameron Martin formula gives rise to an integration by parts formula on E displaystyle E if F E R displaystyle F E to mathbf R has bounded Frechet derivative D F E Lin E R E displaystyle mathrm D F E to operatorname Lin E mathbf R E integrating the Cameron Martin formula with respect to Wiener measure on both sides gives E F x t i h d g x E F x exp t h x 1 2 t 2 h H 2 d g x displaystyle int E F x ti h mathrm d gamma x int E F x exp left t langle h x rangle sim tfrac 1 2 t 2 h H 2 right mathrm d gamma x for any t R displaystyle t in mathbf R Formally differentiating with respect to t displaystyle t and evaluating at t 0 displaystyle t 0 gives the integration by parts formula E D F x i h d g x E F x h x d g x displaystyle int E mathrm D F x i h mathrm d gamma x int E F x langle h x rangle sim mathrm d gamma x Comparison with the divergence theorem of vector calculus suggests d i v V h x h x displaystyle mathop mathrm div V h x langle h x rangle sim where V h E E displaystyle V h E to E is the constant vector field V h x i h displaystyle V h x i h for all x E displaystyle x in E 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 EditUsing Cameron Martin theorem one may establish See Liptser and Shiryayev 1977 p 280 that for a q q displaystyle q times q symmetric non negative definite matrix H t displaystyle H t whose elements H j k t displaystyle H j k t are continuous and satisfy the condition 0 1 j k 1 q H j k t d t lt displaystyle int 0 1 sum j k 1 q H j k t dt lt infty it holds for a q displaystyle q dimensional Wiener process w t displaystyle w t that E exp 0 1 w t H t w t d t exp 1 2 0 1 tr G t d t displaystyle E left exp left int 0 1 w t H t w t dt right right exp left tfrac 1 2 int 0 1 operatorname tr G t dt right where G t displaystyle G t is a q q displaystyle q times q nonpositive definite matrix which is a unique solution of the matrix valued Riccati differential equation d G t d t 2 H t G 2 t displaystyle frac dG t dt 2H t G 2 t See also EditGirsanov theorem theorem Sazonov s theoremReferences EditCameron 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 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