A class of integration by parts formulae in stochastic analysis I

  • K. D. Elworthy
  • Xue-Mei Li


Consider a Stratonovich stochastic differential equation
$$d{{\chi }_{t}}=X\left( {{\chi }_{_{t}}} \right)od{{B}_{t}}+A\left( {{\chi }_{t}} \right)dt$$
with C coefficients on a compact Riemannian manifold M, with associated differential generator \(A=\frac{1}{2}{{\Delta }_{M}}+Z\) and solution flow {ξt : t ≥ 0} of random smooth diffeomorphisms of M. Let Tξt: TM → TM be the induced map on the tangent bundle of M obtained by differentiating ξt with respect to the initial point. Following an observation by A. Thalmaier we extend the basic formula of [EL94] to obtain
$$Edf\left( T\xi .\left( h. \right) \right)=EF\left( \xi .\left( \chi \right) \right)\int_{0}^{T}{\left\langle T{{\xi }_{s}} \right.}\left( {{{\dot{h}}}_{s}} \right),X\left( {{\xi }_{s}}\left( \chi \right) \right)d\left. {{B}_{s}} \right\rangle $$
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 0 2,1 ([0,T];T x M).Set F t x = σ{ξs(x) : 0 ≤ st} Taking conditional expectation with respect to.F T x , formula (1.2) yields integration by parts formulae on C x (M) of the form
$$EdF\left( \gamma \right)\left( {{\overline{V}}^{h}} \right)=EF\left( \gamma \right)\delta {{\overline{V}}^{h}}\left( \gamma \right)$$
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
$$\delta {{\overline{V}}^{h}}(\gamma )=IE\left\{ \int_{0}^{T}{<T{{\xi }_{s}}({{{\dot{h}}}_{s}}),X({{\xi }_{s}}(x))d{{B}_{s}}>|\xi .(x)=\gamma } \right\}$$


Vector Field Stochastic Differential Equation Solution Flow Stochastic Analysis Compact Riemannian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • K. D. Elworthy
    • 1
  • Xue-Mei Li
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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