Cameron–Martin theorem

Statement of the theorem

For abstract Wiener spaces (Banach space setting)

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 ${\displaystyle (T_h{)}_{\ast }(\gamma )}$ is equivalent 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 the Paley–Wiener integral.

The Cameron–Martin formula is valid only for translations by elements of the dense subspace ${\displaystyle i(H)\subseteq E}$, called Cameron–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 finite Borel 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 the trivial (zero) measure. (See quasi-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 as Cameron–Martin directions.

General version for locally convex vector spaces

Consider a locally convex vector space ${\displaystyle E}$, with a Gaussian measure ${\displaystyle \gamma }$ on the cylindrical 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 the Cameron–Martin space ${\displaystyle H_{\gamma }}$ of ${\displaystyle \gamma }$ in ${\displaystyle E}$ is

${\displaystyle H_{\gamma }=\{h\in E:\Vert h{\Vert }_{H_{\gamma }}<\mathrm{\infty }\}}$.

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 is equivalence ${\displaystyle {\gamma }_h\sim \gamma }$ 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 are singular.

Integration by parts

The Cameron–Martin formula gives rise to an integration by parts formula on ${\displaystyle E}$: if ${\displaystyle F:E\to \mathbf{R}}$ has bounded Fréchet 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 the divergence theorem of vector 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 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 ${\displaystyle q\times q}$ symmetric non-negative definite matrix ${\displaystyle H(t)}$ whose elements ${\displaystyle H_{j,k}(t)}$ are continuous and satisfy the condition

${\displaystyle {\int }_0^T\sum _{j,k=1}^q|H_{j,k}(t)|dt<\mathrm{\infty },}$

it holds for a ${\displaystyle q}$−dimensional Wiener 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-valued Riccati 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: $${\displaystyle E\left[\exp \left(-\frac{1}{2}{\int }_0^{T}w(t{)}^{2}dt\right)\right]=\frac{1}{\sqrt{\cosh T}}.}$$