Cameron–Martin formula

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

Recall that standard Gaussian measure γn on n-dimensional Euclidean space Rn is not translation-invariant, but does satisfy the relation

\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),

where the derivative on the left-hand side is the Radon–Nikodym derivative, and (Th)n) is the push forward of the standard Gaussian measure γn by the translation map Th : Rn → Rn, defined by Th(x) = x + h.

Abstract Wiener measure γ on a separable Banach space E, where i : H → 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.

Statement of the theorem

Let i : H → E be an abstract Wiener space with abstract Wiener measure γ : Borel(E) → [0, 1]. For h ∈ H, define Th : E → E by Th(x) = x + i(h). Then (Th)(γ) is equivalent to γ with Radon–Nikodym derivative

\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

\langle h, x \rangle^{\sim} = I(h) (x)

denotes the Paley–Wiener integral.

It is important to note that the Cameron–Martin formula is only valid for translations by elements of the dense subspace i(H) ⊆ 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 μ 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 μ is the trivial (zero) measure. (See quasi-invariant measure.)

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

Proof of Theorem

Integration by parts

The Cameron–Martin formula gives rise to an integration by parts formula on E: if F : E → R has bounded Fréchet derivative DF : E → Lin(ER) = E, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives

\int_{E} F(x + t i(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. Formally differentiating with respect to t and evaluating at t = 0 gives the integration by parts formula

\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

\mathop{\mathrm{div}} [V_h] (x) = - \langle h, x \rangle^\sim,

where Vh : E → E is the constant "vector field" Vh(x) = i(h) for all x ∈ 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 q × q symmetric non-negative definite matrix, H(t) whose elements Hj,k(t) are continuous and satisfy the condition

\int_0^1 \sum_{j,k=1} ^q |H_{j,k}(t)|dt < \infty,

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

E \left[ \exp \left( -\int_0^1 w'(t)H(t)w(t) dt \right) \right] = \exp \left[ \tfrac{1}{2} \int_0^1 tr (G(t)) dt \right],

where G(t) is a q × q nonpositive definite matrix which is a unique solution of the matrix-valued Riccati differential equation

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

See also

References

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