Stochastic flows and Taylor series. (Flots et séries de Taylor stochastiques.) (French) Zbl 0639.60062

Ben Arous, Gérard

Probab. Theory Relat. Fields 81, No. 1, 29-77 (1989).

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

Keywords:

expansion of the solution of a stochastic differential equation; invariant diffusion on a Lie group

Citations:

Zbl 0359.60087; Zbl 0391.60056; Zbl 0427.60060; Zbl 0495.60064; Zbl 0452.60069; Zbl 0438.60047; Zbl 0427.60069

References:

[1]	Azencott, R.; Azema, J.; Yor, M., Formule de Taylor stochastique et développements asymptotiques d’intégrales de Feynmann, Séminaire de probabilités XVI, 237-284 (1982), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[2]	Azencott, R.; Azema, J.; Yor, M., Densités des diffusions en temps petit: developpements asymptotiques, Séminaire de probabilités XVIII, 402-498 (1984), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[3]	Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemanienne (1971), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0223.53034
[4]	Bismut, J. M., Mecanique alétoire (1981), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0457.60002
[5]	Bourbaki, N., Groupes et algèbres de Lie, tome 2 (1972), Paris: Masson, Paris · Zbl 0244.22007
[6]	Doss, H., Lien entre équations différentielles stochastiques et ordinaires, Ann. Inst. Henri Poincare, Novv. Ser., Sect. B, 13, 99-125 (1977) · Zbl 0359.60087
[7]	Fliess, M.; Normand-Cyrot, D.; Azema, J.; Yor, M., Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K. T. Chen, Seminaire de probabilités XVI, 257-267 (1982), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0495.60064
[8]	Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139, 95-153 (1977) · Zbl 0366.22010
[9]	Ibero, M., Intégrales stochastiques multiplicatives, Bull. Sci. Math., 100, 175-191 (1976) · Zbl 0337.60052
[10]	Kunita, H.; Williams, D., On the decomposition of solutions of stochastic differential equations, Proceedings, LMS Durham Symposium, 1980. Lect. Notes Math., vol. 851, 213-284 (1981), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0474.60046
[11]	Kunita, H.; Azema, J.; Yor, M., On the representation of solutions of stochastic differential equations, Séminaire de probabilités XIV, 282-304 (1980), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0438.60047
[12]	Malliavin, P.: Parametrix trajectorielle pour un opérateur hypoelliptique et repère mobile stochastique. C.R. Acad. Sci., Paris, Ser. I 281, 241 (1975) · Zbl 0325.60056
[13]	Malliavin, P., Géométrie différentielle stochastique (1978), Montréal: Presses de l’université de Montréal, Montréal · Zbl 0393.60062
[14]	Meyer, P. A.; Meyer, P. A., Cours sur l’intégrale stochastique, Séminaire de probabilités X, 321-331 (1976), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[15]	Palais, R., A global formulation of the Lie theory on transformation groups, Mem. Am. Math. Soc., 22, 95-97 (1957) · Zbl 0178.26502
[16]	Platen, E., A Taylor formula for semimartingales solving a stochastic equation, Third conference on stochastic differential systems, 65-68 (1980), Visegrad: Hongrie, Visegrad
[17]	Sussmann, H., On the gap between deterministic and stochastic ordinary equations, Ann. Probab., 6, 19-41 (1978) · Zbl 0391.60056
[18]	Yamato, Y., Stochastic differential equations and nilpotent Lie algebras, Z. Wahrscheinlichkeitstheor. Verw. Geb., 47, 213-229 (1979) · Zbl 0427.60069
[19]	Krener, A. J.; Lobry, C., The complexity of stochastic differential equations, Stochastics, 4, 193-203 (1981) · Zbl 0452.60069
[20]	Abraham, R.; Marsden, J.; Ratiu, T., Manifolds, tensor analysis and applications (1983), Reading, Mass.: Addison Wesley, Reading, Mass. · Zbl 0508.58001
[21]	Nagano, T., Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan, 18, 398-404 (1966) · Zbl 0147.23502