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Geometry of stochastic delay differential equations. (English) Zbl 1111.60041

Summary: Stochastic delay differential equations (SDDE) on a manifold \(M\) depend intrinsically on a connection \(\nabla\) in this space. The main geometric result in this note concerns the horizontal lift of solutions of SDDE on a manifold \(M\) to an SDDE in the frame bundle \(BM\), hence the lifted equation should come together with the prolonged horizontal connection \(\nabla^H\) on \(BM\). We show that every horizontal semimartingale can be represented as a solution of an SDDE.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K50 Stochastic functional-differential equations
58J65 Diffusion processes and stochastic analysis on manifolds
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