Asymptotic expansion of stochastic flows
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 \textit{H. Doss} [Ann. Inst. H. Poincaré, n. Sér., Sect. B 13, 99-125 (1977; Zbl 0359.60087)], \textit{H. J. Sussmann} [Ann. Probab. 6, 19-41 (1978; Zbl 0391.60056)], \textit{M. Fliess} and \textit{D. Normand-Cyrot} [Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Lect. Notes Math. 920, 257-267 (1982; Zbl 0495.60064)], \textit{A. J. Krener} and \textit{C. Lobry} [Stochastics 4, 193- 203 (1981; Zbl 0452.60069)], \textit{Y. Yamato} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 47, 213-229 (1979; Zbl 0427.60069)] and \textit{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 \textit{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 \textit{R. S. Strichartz} [J. Funct. Anal. 72, 320-345 (1987; Zbl 0623.34058)].
iterated stochastic Stratonovich integrals
deterministic ordinary differential equations
scientific article; zbMATH DE number 569089
scientific article