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Asymptotic expansion of stochastic flows. (English) Zbl 0794.60054
We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the reslts of H. Doss [Ann. Inst. H. Poincaré, n. Sér., Sect. B 13, 99-125 (1977; Zbl 0359.60087)], H. J. Sussmann [Ann. Probab. 6, 19-41 (1978; Zbl 0391.60056)], 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)], Y. Yamato [Z. Wahrscheinlichkeitstheorie Verw. Geb. 47, 213-229 (1979; Zbl 0427.60069)] and H. Kunita [Séminaire de probabilités XIV, 1978/79, Lect. Notes Math. 784, 282-304 (1980; Zbl 0438.60047)] in the nilpotent case, and extends to general diffusions the representation given by G. Ben Arous [Probab. Theory Relat. Fields 81, No. 1, 29-77 (1989; Zbl 0639.60062)] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by R. S. Strichartz [J. Funct. Anal. 72, 320-345 (1987; Zbl 0623.34058)].
Reviewer: F.Castell (Orsay)

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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