Inmathematics, theCameron–Martin theorem orCameron–Martin formula (named afterRobert Horton Cameron andW. T. Martin) is atheorem ofmeasure theory that describes howabstract Wiener measure changes undertranslation by certain elements of the Cameron–MartinHilbert space.

The standardGaussian measure${\displaystyle {\gamma }^n}$ on${\displaystyle n}$-dimensionalEuclidean space${\displaystyle {\mathbf{R}}^n}$ is not translation-invariant. (In fact, there is a unique translation invariantRadon measure up to scale byHaar's theorem: the${\displaystyle n}$-dimensionalLebesgue measure, denoted here${\displaystyle dx}$.) Instead, a measurable subset${\displaystyle A}$ hasGaussian measure

${\displaystyle {\gamma }_n(A)=\frac{1}{(2\pi {)}^{n/2}}{\int }_A\exp \left(-\frac{1}{2}⟨x,x{⟩}_{{\mathbf{R}}^n}\right)dx.}$

Here${\displaystyle ⟨x,x{⟩}_{{\mathbf{R}}^n}}$ refers to the standard Euclideandot product in${\displaystyle {\mathbf{R}}^n}$. The Gaussian measure of the translation of${\displaystyle A}$ by a vector${\displaystyle h\in {\mathbf{R}}^n}$ is

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

${\displaystyle \exp \left(\frac{2⟨x,h{⟩}_{{\mathbf{R}}^n}-⟨h,h{⟩}_{{\mathbf{R}}^n}}{2}\right)=\exp \left(⟨x,h{⟩}_{{\mathbf{R}}^n}-\frac{1}{2}\Vert h{\Vert }_{{\mathbf{R}}^n}^2\right).}$

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

${\displaystyle \frac{\mathrm{d}(T_h{)}_{\ast }({\gamma }^n)}{\mathrm{d}{\gamma }^n}(x)=\exp \left({⟨h,x⟩}_{{\mathbf{R}}^n}-\frac{1}{2}\Vert h{\Vert }_{{\mathbf{R}}^n}^2\right).}$

The abstract Wiener measure${\displaystyle \gamma }$ on aseparableBanach space${\displaystyle E}$, where${\displaystyle i:H\to E}$ is anabstract 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 thedensesubspace${\displaystyle i(H)\subseteq E}$.

Let${\displaystyle i:H\to E}$ be an abstract Wiener space with abstract Wiener measure${\displaystyle \gamma :\mathrm{Borel}(E)\to [0,1]}$. For${\displaystyle h\in H}$, define${\displaystyle T_h:E\to E}$ by${\displaystyle T_h(x)=x+i(h)}$. Then${\displaystyle (T_h{)}_{\ast }(\gamma )}$ isequivalent to${\displaystyle \gamma }$ with Radon–Nikodym derivative

${\displaystyle \frac{\mathrm{d}(T_h{)}_{\ast }(\gamma )}{\mathrm{d}\gamma }(x)=\exp \left(⟨h,x{⟩}^{\sim }-\frac{1}{2}\Vert h{\Vert }_H^2\right),}$

where

${\displaystyle ⟨h,x{⟩}^{\sim }=i(h)(x)}$

denotes thePaley–Wiener integral.

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

If${\displaystyle E}$ is a separable Banach space and${\displaystyle \mu }$ is a locally finiteBorel measure on${\displaystyle E}$ that is equivalent to its own push forward under any translation, then either${\displaystyle E}$ has finite dimension or${\displaystyle \mu }$ is thetrivial (zero) measure. (Seequasi-invariant measure.)

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

Consider alocally convex vector space${\displaystyle E}$, with a Gaussian measure${\displaystyle \gamma }$ on thecylindrical σ-algebra${\displaystyle \sigma (\mathrm{Cyl}(E,E^{\prime }))}$ and let${\displaystyle {\gamma }_m:=\gamma (\cdot -m)}$ denote the translation by${\displaystyle m\in E}$. For an element in the topological dual${\displaystyle f\in E^{\prime }}$ define the distance to the mean${\displaystyle t_{\gamma }(f):=f-{\mathbb{E}}_{\gamma }[f],}$and denote the closure in${\displaystyle L^2(E,\gamma )}$ as${\displaystyle E_a^{\gamma }:=\mathrm{clos}\left\{(t_{\gamma }(f_n){)}_n:f\in E^{\prime }\right\}}$.Define the covariance operator${\displaystyle \overline{R_{\gamma }}:E_a^{\gamma }\to (E^{\prime }{)}^{\ast }}$ extended to the closure as

${\displaystyle \overline{R_{\gamma }}(f)(g)=⟨f,g-{\mathbb{E}}_{\gamma }[g]{⟩}_{L^2(\gamma )}}$.

Define the norm

${\displaystyle \Vert h{\Vert }_{H_{\gamma }}:=sup\{f(h):f\in E^{\prime },\overline{R_{\gamma }}(f)(f)\le 1\},}$

then theCameron–Martin space${\displaystyle H_{\gamma }}$ of${\displaystyle \gamma }$ in${\displaystyle E}$ is

.

If for${\displaystyle h\in E}$ there exists an${\displaystyle g\in E_a^{\gamma }}$ such that${\displaystyle h=\overline{R_{\gamma }}(g)}$ then${\displaystyle h\in H_{\gamma }}$ and${\displaystyle \Vert h{\Vert }_{H_{\gamma }}=\Vert g{\Vert }_{L^2(\gamma )}}$. Further there isequivalence${\displaystyle {\gamma }_h\sim \gamma }$ with Radon-Nikodým density

${\displaystyle \frac{d{\gamma }_h}{d\gamma }=\exp \left(g(x)-\frac{1}{2}\Vert h{\Vert }_{H_{\gamma }}^2\right).}$

If${\displaystyle h\notin H_{\gamma }}$ the two measures aresingular.^[1]

The Cameron–Martin formula gives rise to anintegration by parts formula on${\displaystyle E}$: if${\displaystyle F:E\to \mathbf{R}}$ hasboundedFréchet derivative${\displaystyle \mathrm{D}F:E\to \mathrm{Lin}(E;\mathbf{R})=E^{\ast }}$, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives

${\displaystyle {\int }_{E}F(x+ti(h))\mathrm{d}\gamma (x)={\int }_{E}F(x)\exp \left(t⟨h,x{⟩}^{\sim }-\frac{1}{2}{t}^2\Vert h{\Vert }_H^2\right)\mathrm{d}\gamma (x)}$

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

${\displaystyle {\int }_E\mathrm{D}F(x)(i(h))\mathrm{d}\gamma (x)={\int }_{E}F(x)⟨h,x{⟩}^{\sim }\mathrm{d}\gamma (x).}$

Comparison with thedivergence theorem ofvector calculus suggests

${\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}[V_h](x)=-⟨h,x{⟩}^{\sim },}$

where${\displaystyle V_h:E\to E}$ is the constant "vector field"${\displaystyle V_h(x)=i(h)}$ for all${\displaystyle x\in E}$. The wish to consider more general vector fields and to think of stochastic integrals as "divergences" leads to the study ofstochastic processes and theMalliavin calculus, and, in particular, theClark–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${\displaystyle q\times q}$ symmetric non-negativedefinite matrix${\displaystyle H(t)}$ whose elements${\displaystyle H_{j,k}(t)}$ are continuous and satisfy the condition

it holds for a${\displaystyle q}$−dimensionalWiener process${\displaystyle w(t)}$ that

${\displaystyle E\left[\exp \left(-{\int }_0^{T}w(t{)}^{\ast }H(t)w(t)dt\right)\right]=\exp \left[\frac{1}{2}{\int }_0^T\mathrm{tr}(G(t))dt\right],}$

where${\displaystyle G(t)}$ is a${\displaystyle q\times q}$ nonpositive definite matrix which is a unique solution of the matrix-valuedRiccati differential equation

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

with the boundary condition${\displaystyle G(T)=0}$.

In the special case of a one-dimensional Brownian motion where${\displaystyle H(t)=1/2}$, the unique solution is${\displaystyle G(t)=\tanh (t-T)}$, and we have the original formula as established by Cameron and Martin:$${\displaystyle E\left[\exp \left(-\frac{1}{2}{\int }_0^{T}w(t{)}^{2}dt\right)\right]=\frac{1}{\sqrt{\cosh T}}.}$$

• Girsanov theorem – Theorem on changes in stochastic processes
• Sazonov's theorem

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