Summary
We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) 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 Doss [6], Sussmann [17], Yamato [18], Fliess and Normand-Cyrot [7], Krener and Lobry [19] and Kunita [11] on the representation of solutions of stochastic differential equations.
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Azencott, R.: Formule de Taylor stochastique et développements asymptotiques d'intégrales de Feynmann. In: Azema, J., Yor, M. (eds.) Séminaire de probabilités XVI (Lect. Notes Math., vol. 921, pp. 237–284) Berlin Heidelberg New York: Springer 1982
Azencott, R.: Densités des diffusions en temps petit: developpements asymptotiques. In: Séminaire de probabilités XVIII. (Azema, J., Yor, M. (eds.) (Lect. Notes Math., vol. 1059, pp. 402–498). Berlin Heidelberg New York: Springer 1984
Berger, M., Gauduchon, P., Mazet, E.: Le spectre d'une variété riemanienne. (Lect. Notes Math., vol. 194). Berlin Heidelberg New York: Springer 1971
Bismut, J.M.: Mecanique alétoire. (Lect. Notes Math., vol. 866). Berlin Heidelberg New York: Springer 1981
Bourbaki, N.: Groupes et algèbres de Lie, tome 2. Paris: Masson 1972
Doss, H.: Lien entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincare, Novv. Ser., Sect. B 13, 99–125 (1977)
Fliess, M., Normand-Cyrot, D.: Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K. T. Chen. In: Azema, J., Yor, M. (eds.) Seminaire de probabilités XVI (Lect. Notes Math., vol. 920, pp. 257–267). Berlin Heidelberg New York: Springer 1982
Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139, 95–153 (1977)
Ibero, M.: Intégrales stochastiques multiplicatives. Bull. Sci. Math. 100, 175–191 (1976)
Kunita, H.: On the decomposition of solutions of stochastic differential equations. In: Williams, D. (red.) Proceedings, LMS Durham Symposium, 1980. Lect. Notes Math., vol. 851, pp. 213–284). Berlin Heidelberg New York: Springer 1981
Kunita, H.: On the representation of solutions of stochastic differential equations. Séminaire de probabilités XIV. In: Azema, J., Yor; M. (eds.), (Lect. Notes Math., vol. 784, pp. 282–304. Berlin Heidelberg New York: Springer 1980
Malliavin, P.: Parametrix trajectorielle pour un opérateur hypoelliptique et repère mobile stochastique. C.R. Acad. Sci., Paris, Ser. I 281, 241 (1975)
Malliavin, P.: Géométrie différentielle stochastique. Montréal: Presses de l'université de Montréal 1978
Meyer, P.A.: Cours sur l'intégrale stochastique. In: Meyer, P.A. (ed.) Séminaire de probabilités X. (Lect. Notes Math., vol. 511, pp. 321–331). Berlin Heidelberg New York: Springer 1976
Palais, R.: A global formulation of the Lie theory on transformation groups. Mem. Am. Math. Soc. 22, 95–97 (1957)
Platen, E.: A Taylor formula for semimartingales solving a stochastic equation. In: Third conference on stochastic differential systems, pp. 65–68. Visegrad: Hongrie 1980
Sussmann, H.: On the gap between deterministic and stochastic ordinary equations. Ann. Probab. 6, 19–41 (1978)
Yamato, Y.: Stochastic differential equations and nilpotent Lie algebras. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 213–229 (1979)
Krener, A.J., Lobry, C.: The complexity of stochastic differential equations. Stochastics 4, 193–203 (1981)
Abraham, R., Marsden, J., Ratiu, T.: Manifolds, tensor analysis and applications. Reading, Mass.: Addison Wesley 1983
Nagano, T.: Linear differential systems with singularities and applications to transitive Lie algebras. J. Math. Soc. Japan 18, 398–404 (1966)
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Arous, G.B. Flots et series de Taylor stochastiques. Probab. Th. Rel. Fields 81, 29–77 (1989). https://doi.org/10.1007/BF00343737
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DOI: https://doi.org/10.1007/BF00343737