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Regularization of the Stratonovich equations with jumps between manifolds. (Régularisation d’équations de Stratonovitch à sauts entre variétés.) (French) Zbl 0922.58086

From the authors’ abstract: We investigate a way to regularize a general manifold valued semimartingale, in order to approximate the solution of some stochastic differential equation. A regularization of the stochastic driving noise is used. The regularization procedure is the following: a sample path of the cadlag manifold valued semimartingale is fixed, and the semimartingale position at time \(t\) is approximated by a “mean value” of the sample path just before time \(t\). The usual definition of vector-valued barycenter has to be extended to construct this intrinsic “mean value” , and the resulting regularized processes, which are continuous with finite variation, produce an estimate of the original semimartingale. The solutions of the ordinary differential equations driven by those finite variation processes are proved to converge to the so-called solutions of stochastic differential equations between manifolds.
The results obtained are a natural geometrical extension of a paper by Kurtz, Pardoux and Protter. The geometric point of view allows us to deal with a discontinuous driving process even if the regularization is not linear.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60J75 Jump processes (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
60H20 Stochastic integral equations
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References:

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