Theorem of measure theory
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
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
![{\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773857977227073724b648717f31cae09f4adb38)
Here
refers to the standard Euclidean dot product in
. The Gaussian measure of the translation of
by a vector
is
![{\displaystyle {\begin{aligned}\gamma _{n}(A-h)&={\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]&={\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7a16b88418d7310a64e0160bd3ee36b852224b)
So under translation through
, the Gaussian measure scales by the distribution function appearing in the last display:
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa0d8167c991172139b76fd02134aabe6e866f4)
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
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d70cc249b55c25fddb9f1d8e466e714d17621e)
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
Let
be an abstract Wiener space with abstract Wiener measure
. For
, define
by
. Then
is equivalent to
with Radon–Nikodym derivative
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbabb1ed28d9cdb0827b4fa17490bb5aba2c8496)
where
![{\displaystyle \langle h,x\rangle ^{\sim }=i(h)(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4609e675a061588081be76303809c323222ad7)
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
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
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d466a68b1a818f1851610f06c9224f4ffa79dc66)
for any
. Formally differentiating with respect to
and evaluating at
gives the integration by parts formula
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1192a5a1568c4a284ad802b0f223d9515a528223)
Comparison with the divergence theorem of vector calculus suggests
![{\displaystyle \mathop {\mathrm {div} } [V_{h}](x)=-\langle h,x\rangle ^{\sim },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf72ced5cec79a1e11d718bb03efbc222fa1347)
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
![{\displaystyle \int _{0}^{1}\sum _{j,k=1}^{q}|H_{j,k}(t)|\,dt<\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7b046cd7b7aeecdb5391462a1ebe5400bf737c)
it holds for a
−dimensional Wiener process
that
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e18615c3377ea4f26350e4e45ac36bd60b4c2511)
where
is a
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cc03f106be40264af4bdc2759141b6c42575ad)
See also
References
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