A class of integration by parts formulae in stochastic analysis I

Abstract

Consider a Stratonovich stochastic differential equation

$$d{{\chi }_{t}}=X\left( {{\chi }_{_{t}}} \right)od{{B}_{t}}+A\left( {{\chi }_{t}} \right)dt$$

(1.1)

with C^∞ coefficients on a compact Riemannian manifold M, with associated differential generator $A=\frac{1}{2}{{\Delta }_{M}}+Z$ and solution flow {ξ_t : t ≥ 0} of random smooth diffeomorphisms of M. Let Tξ_t: TM → TM be the induced map on the tangent bundle of M obtained by differentiating ξ_t with respect to the initial point. Following an observation by A. Thalmaier we extend the basic formula of [EL94] to obtain

$$Edf\left( T\xi .\left( h. \right) \right)=EF\left( \xi .\left( \chi \right) \right)\int_{0}^{T}{\left\langle T{{\xi }_{s}} \right.}\left( {{{\dot{h}}}_{s}} \right),X\left( {{\xi }_{s}}\left( \chi \right) \right)d\left. {{B}_{s}} \right\rangle $$

(1.2)

where $F\in FC_{b}^{\infty }\left( {{C}_{\chi }}\left( M \right) \right)$, the space of smooth cylindrical functions on the space C _x (M) of continuous paths γ : [0,T] → M with γ(0) = x, dF is its derivative, and h. is a suitable adapted process with sample paths in the Cameron-Martin space L ^2,1₀ ([0,T];T _x M).Set F ^x_t = σ{ξ_s(x) : 0 ≤ s ≤ t} Taking conditional expectation with respect to.F ^x_T , formula (1.2) yields integration by parts formulae on C _x (M) of the form

$$EdF\left( \gamma \right)\left( {{\overline{V}}^{h}} \right)=EF\left( \gamma \right)\delta {{\overline{V}}^{h}}\left( \gamma \right)$$

(1.3)

where ${{\overline{V}}^{h}}$ is the vector field on C _x(M)

$${{\overline{V}}^{h}}{{\left( \gamma \right)}_{t}}-E\left\{ T{{\xi }_{t}}\left( {{h}_{t}} \right)\left| \xi .\left( \chi \right)=\gamma \right. \right\}$$

and $\delta {{\overline{V}}^{h}}:{{C}_{x}}\left( M \right)\to$ is given by

$$\delta {{\overline{V}}^{h}}(\gamma )=IE\left\{ \int_{0}^{T}{<T{{\xi }_{s}}({{{\dot{h}}}_{s}}),X({{\xi }_{s}}(x))d{{B}_{s}}>|\xi .(x)=\gamma } \right\}$$

.