×

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
PDF BibTeX XML Cite
Full Text: DOI

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
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.