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Stochastic flows and Taylor series. (Flots et séries de Taylor stochastiques.) (French) Zbl 0639.60062
We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovich) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the Brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor.
The first of these formulae contains, and extends to the non nilpotent case, the results of H. Doss [Ann. Inst. Henri Poincaré, n. Sér., Sect. B 13, 99–125 (1977; Zbl 0359.60087)], H. Sussmann [Ann. Probab. 6, 19–41 (1978; Zbl 0391.60056)], Y. Yamato [Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 213–229 (1979; Zbl 0427.60069)], M. Fliess and D. Normand-Cyrot [Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Lect. Notes Math. 920, 257–267 (1982; Zbl 0495.60064)], A. J. Krener and C. Lobry [Stochastics 4, 193–203 (1981; Zbl 0452.60069)] and H. Kunita [Séminaire de probabilités XIV, 1978/79, Lect. Notes Math. 784, 282–304 (1980; Zbl 0438.60047)] on the representation of solutions of stochastic differential equations.
Reviewer: G.Ben Arous

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F99 Limit theorems in probability theory
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