Cameron–Martin theorem

The standard Gaussian measure $\backslash gamma^n$ on $n$-dimensional Euclidean space $\backslash mathbf\{R\}^n$ is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the $n$-dimensional Lebesgue measure, denoted here $dx$.) Instead, a measurable subset $A$ has Gaussian measure

:$\backslash gamma\_n(A)\; =\; \backslash frac\{1\}\{(2\backslash pi)^\{n/2\}\}\backslash int\_A\; \backslash exp\backslash left(-\backslash tfrac12\backslash langle\; x,\; x\backslash rangle\_\{\backslash mathbf\; R^n\}\backslash right)\backslash ,dx.$

Here $\backslash langle\; x,x\backslash rangle\_\{\backslash mathbf\; R^n\}$ refers to the standard Euclidean dot product in $\backslash mathbf\{R\}^n$. The Gaussian measure of the translation of $A$ by a vector $h\; \backslash in\; \backslash mathbf\{R\}^n$ is

:$\backslash begin\{align\}$

\gamma_n(A-h) &= \frac{1}{(2\pi)^{n/2}}\int_A \exp\left(-\tfrac12\langle x-h, x-h\rangle_{\mathbf R^n}\right)\,dx\\[4pt]

&=\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(-\tfrac12\langle x, x\rangle_{\mathbf R^n}\right)\,dx.

\end{align}

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

:$\backslash exp\backslash left(\backslash frac\{2\backslash langle\; x,\; h\backslash rangle\_\{\backslash mathbf\; R^n\}\; -\; \backslash langle\; h,\; h\backslash rangle\_\{\backslash mathbf\; R^n\}\}\{2\}\; \backslash right)=\backslash exp\backslash left(\backslash langle\; x,\; h\backslash rangle\_\{\backslash mathbf\; R^n\}\; -\; \backslash tfrac12\backslash |h\backslash |\_\{\backslash mathbf\; R^n\}^\; 2\backslash right).$

The measure that associates to the set $A$ the number $\backslash gamma\_n(A\; -\; h)$ is the pushforward measure, denoted $(T\_h)\_*\; (\backslash gamma^n).$ Here $T\_h\; :\; \backslash mathbf\{R\}^n\; \backslash to\; \backslash mathbf\{R\}^n$ refers to the translation map: $T\_h(x)\; =\; x\; +\; h$. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by

:$\backslash frac\{\backslash mathrm\{d\}\; (T\_h)\_\{*\}\; (\backslash gamma^n)\}\{\backslash mathrm\{d\}\; \backslash gamma^n\}\; (x)\; =\; \backslash exp\; \backslash left(\; \backslash left\; \backslash langle\; h,\; x\; \backslash right\; \backslash rangle\_\{\backslash mathbf\{R\}^n\}\; -\; \backslash tfrac\{1\}\{2\}\; \backslash |\; h\; \backslash |\_\{\backslash mathbf\{R\}^n\}^2\; \backslash right).$

The abstract Wiener measure $\backslash gamma$ on a separable Banach space $E$, where $i\; :\; H\; \backslash to\; E$ 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 $i(H)\; \backslash subseteq\; E$.

Let $i\; :\; H\; \backslash to\; E$ be an abstract Wiener space with abstract Wiener measure $\backslash gamma\; :\; \backslash operatorname\{Borel\}(E)\; \backslash to\; [0,\; 1]$. For $h\; \backslash in\; H$, define $T\_h\; :\; E\; \backslash to\; E$ by $T\_h(x)\; =\; x\; +\; i(h)$. Then $(T\_h)\_*(\backslash gamma)$ is equivalent to $\backslash gamma$ with Radon–Nikodym derivative

:$\backslash frac\{\backslash mathrm\{d\}\; (T\_\{h\})\_\{*\}\; (\backslash gamma)\}\{\backslash mathrm\{d\}\; \backslash gamma\}\; (x)\; =\; \backslash exp\; \backslash left(\; \backslash langle\; h,\; x\; \backslash rangle^\{\backslash sim\}\; -\; \backslash tfrac\{1\}\{2\}\; \backslash |\; h\; \backslash |\_\{H\}^\{2\}\; \backslash right),$

where

:$\backslash langle\; h,\; x\; \backslash rangle^\{\backslash sim\}\; =\; i(h)\; (x)$

denotes the Paley–Wiener integral.

The Cameron–Martin formula is valid only for translations by elements of the dense subspace $i(H)\; \backslash subseteq\; E$, called Cameron–Martin space, and not by arbitrary elements of $E$. If the Cameron–Martin formula did hold for arbitrary translations, it would contradict the following result:

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

In fact, $\backslash gamma$ is quasi-invariant under translation by an element $v$ if and only if $v\; \backslash in\; i(H)$. Vectors in $i(H)$ are sometimes known as Cameron–Martin directions.

Consider a locally convex vector space $E$, with a Gaussian measure $\backslash gamma$ on the cylindrical σ-algebra $\backslash sigma(\backslash operatorname\{Cyl\}(E,E\text{'}))$ and let $\backslash gamma\_\{m\}:=\backslash gamma(\backslash cdot\; -\; m)$ denote the translation by $m\backslash in\; E$. For an element in the topological dual $f\backslash in\; E\text{'}$ define the distance to the mean $t\_\{\backslash gamma\}(f):=f-\backslash mathbb\{E\}\_\{\backslash gamma\}[f],$

and denote the closure in $L^2(E,\backslash gamma)$ as $E\_a^\{\backslash gamma\}:=\backslash operatorname\{clos\}\backslash left\backslash \{(t\_\{\backslash gamma\}(f\_n))\_\{n\}\backslash colon\backslash \; f\backslash in\; E\text{'}\backslash right\backslash \}$.

Define the covariance operator $\backslash overline\{R\_\{\backslash gamma\}\}:E\_a^\{\backslash gamma\}\backslash to\; (E\text{'})^*$ extended to the closure as

:$\backslash overline\{R\_\{\backslash gamma\}\}(f)(g)=\backslash langle\; f,g-\backslash mathbb\{E\}\_\{\backslash gamma\}[g]\backslash rangle\_\{L^2(\backslash gamma)\}$.

Define the norm

:$\backslash |h\backslash |\_\{H\_\backslash gamma\}:=\backslash sup\backslash \{f(h)\backslash colon\; f\backslash in\; E\text{'},\backslash ;\; \backslash overline\{R\_\{\backslash gamma\}\}(f)(f)\; \backslash leq\; 1\backslash \},$

then the Cameron-Martin space $H\_\backslash gamma$ of $\backslash gamma$ in $E$ is

:.

If for $h\backslash in\; E$ there exists an $g\backslash in\; E\_a^\{\backslash gamma\}$ such that $h=\backslash overline\{R\_\{\backslash gamma\}\}(g)$ then $h\backslash in\; H\_\backslash gamma$ and $\backslash |h\backslash |\_\{H\_\backslash gamma\}\; =\; \backslash |g\backslash |\_\{L^2(\backslash gamma)\}$. Further there is equivalence $\backslash gamma\_h\; \backslash sim\; \backslash gamma$ with Radon-Nikodým density

::$\backslash frac\{d\backslash gamma\_h\}\{d\backslash gamma\}=\backslash exp\backslash left(g(x)-\backslash frac\{1\}\{2\}\backslash |h\backslash |\_\{H\_\backslash gamma\}^2\backslash right).$

If $h\backslash not\backslash in\; H\_\backslash gamma$ the two measures are singular.{{cite book |first1=Vladimir |last1=Bogachev| publisher=American Mathematical Society|title=Gaussian Measures|place=Rhode Island|date=1998}}

The Cameron–Martin formula gives rise to an integration by parts formula on $E$: if $F\; :\; E\; \backslash to\; \backslash mathbf\{R\}$ has bounded Fréchet derivative $\backslash mathrm\{D\}F\; :\; E\; \backslash to\; \backslash operatorname\{Lin\}(E;\; \backslash mathbf\{R\})\; =\; E^*$, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives

:$\backslash int\_\{E\}\; F(x\; +\; t\; i(h))\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x)\; =\; \backslash int\_\{E\}\; F(x)\; \backslash exp\; \backslash left(\; t\; \backslash langle\; h,\; x\; \backslash rangle^\{\backslash sim\}\; -\; \backslash tfrac\{1\}\{2\}\; t^2\; \backslash |\; h\; \backslash |\_\{H\}^\{2\}\; \backslash right)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x)$

for any $t\; \backslash in\; \backslash mathbf\{R\}$. Formally differentiating with respect to $t$ and evaluating at $t\; =\; 0$ gives the integration by parts formula

:$\backslash int\_E\; \backslash mathrm\{D\}\; F(x)\; (i(h))\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x)\; =\; \backslash int\_E\; F(x)\; \backslash langle\; h,\; x\; \backslash rangle^\backslash sim\; \backslash ,\; \backslash mathrm\{d\}\; \backslash gamma\; (x).$

Comparison with the divergence theorem of vector calculus suggests

:$\backslash mathop\{\backslash mathrm\{div\}\}\; [V\_h]\; (x)\; =\; -\; \backslash langle\; h,\; x\; \backslash rangle^\backslash sim,$

where $V\_h\; :\; E\; \backslash to\; E$ is the constant "vector field" $V\_h(x)\; =\; i(h)$ for all $x\; \backslash in\; E$. 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.

Using Cameron–Martin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a $q\; \backslash times\; q$ symmetric non-negative definite matrix $H(t)$ whose elements $H\_\{j,\; k\}(t)$ are continuous and satisfy the condition

:

it holds for a $q$−dimensional Wiener process $w(t)$ that

:$E\; \backslash left[\; \backslash exp\; \backslash left(\; -\backslash int\_0^T\; w(t)^*H(t)w(t)\; \backslash ,\; dt\; \backslash right)\; \backslash right]\; =\; \backslash exp\; \backslash left[\; \backslash tfrac\{1\}\{2\}\; \backslash int\_0^T\; \backslash operatorname\{tr\}\; (G(t))\; \backslash ,\; dt\; \backslash right],$

where $G(t)$ is a $q\; \backslash times\; q$ nonpositive definite matrix which is a unique solution of the matrix-valued Riccati differential equation

:$\backslash frac\{dG(t)\}\{dt\}\; =\; 2H(t)-G^2(t)$

with the boundary condition $G(T)\; =\; 0$.

In the special case of a one-dimensional Brownian motion where $H(t)=1/2$, the unique solution is $G(t)=\backslash tanh(t-T)$, and we have the original formula as established by Cameron and Martin:

$$E\backslash left[\backslash exp\backslash left(-\backslash tfrac12\backslash int\_0^T\; w(t)^2\backslash ,dt\backslash right)\backslash right]\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash cosh\; T\}\}.$$

• {{annotated link|Girsanov theorem}}
• {{annotated link|Sazonov's theorem}}

{{refbegin}}

• {{cite journal|last1=Cameron|first1=R. H.|last2 = Martin|first2 = W. T.|year=1944|title=Transformations of Wiener Integrals under Translations|journal=Annals of Mathematics |volume=45|issue=2|pages=386–396|jstor=1969276|doi=10.2307/1969276}}
• {{cite book

| last1 = Liptser | first1 = R. S.

| last2 = Shiryayev | first2 = A. N.

| title = Statistics of Random Processes I: General Theory

| year = 1977

| publisher = Springer-Verlag

| isbn = 3-540-90226-0

}}

• {{citation

| last1=Elworthy | first1=David

| date=2008

| url=http://www.tjsullivan.org.uk/pdf/MA482_Stochastic_Analysis.pdf

| title=MA482 Stochastic Analysis

| publisher=Lecture Notes, University of Warwick}}

• {{citation

| first1=Alessandra |last1 =Lunardi

| first2 = Michele |last2 = Miranda

| first3 = Diego |last3 = Pallara

| date= 2016

| url=http://www.dm.unife.it/it/ricerca-dmi/seminari/isem19/lectures/lecture-notes/at_download/file

| title=Infinite Dimensional Analysis

| publisher=Lecture Notes, 19th Internet Seminar, Dipartimento di Matematica e Informatica Università degli Studi di Ferrara}}

{{refend}}

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