Summary
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 results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].
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Azencott, R.: Formule de Taylor stochastique et développements asymptotiques d'intégrales de Feynman. 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: développements asymptotiques. In: Azema, J., Yor, M. (eds.) Séminaire de probabilités XVIII. (Lect. Notes Math., vol. 1059, pp. 402–498) Berlin Heidelberg New York: Springer 1984
Ben Arous, G.: Flots et séries de Taylor stochastiques. Probab. Theory Relat. Fields81, 29–77 (1989)
Ben Arous, G.: Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier39, 73–99 (1989)
Bourbaki, N.: Eléments de mathématiques. Groupes et algèbres de Lie, tome 2. Paris: Hermann 1972
Doss, H.: Lien entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. B13, 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.) Séminaire de probabilités XVI. (Lect. Notes Math., vol. 920, pp. 257–267) Berlin Heidelberg New York: Springer 1982
Hu, Y.Z.: Série de Taylor stochastique et formule de Campbell-Hausdorff, d'après Ben Arous. In: Azema, J., Meyer, P.A., Yor, M. (eds.) Séminaire de probabilités XXVI, 1991–1992 (Lect. Notes Math., vol. 1526, pp. 579–586) Berlin Heidelberg New York: Springer 1992
Jacobson, N.: Lie algebras. New York: Interscience 1962
Krener, A.J., Lobry, C.: The complexity of stochastic differential equations. Stochastics4, 193–203 (1979)
Kunita, H.: On the representation of solutions of stochastic differential equations. In: Azema, J., Yor, M. (eds.) Séminaire de probabilités XIV. (Lect. Notes Math., vol. 784, pp. 282–304) Berlin Heidelberg New York: Springer 1980
Kunita, H.: On the decomposition of solutions of stochastic differential equations. In: Williams, D. (ed.) Stochastic integrals. Proceedings, LMS Durham Symposium 1980. (Lect. Notes Math., vol. 851, pp. 213–255) Berlin Heidelberg New York: Springer 1981
Léandre, R.: Applications quantitatives et géométriques du calcul de Malliavin. In: Métivier, M., Watanabe, S. (eds.) Stochastics integrals. (Lect. Notes Math., vol. 1322, pp. 109–133) Berlin Heidelberg New York: Springer 1988
Strichartz, R.S.: The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal.72, 320–345 (1987)
Sussmann, H.: On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab.6, 19–41 (1978)
Takanobu, S.: Diagonal short time asymptotics of heat kernels for certain degenerate second order differential operators of Hörmander type. Publ. Res. Inst. Math. Sci.24, 169–203 (1988)
Yamato, Y.: Stochastic differential equations and nilpotent Lie algebras. Z. Wahrscheinlichkeitstheor. Verw. Geb.47, 213–229 (1979)
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Castell, F. Asymptotic expansion of stochastic flows. Probab. Th. Rel. Fields 96, 225–239 (1993). https://doi.org/10.1007/BF01192134
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DOI: https://doi.org/10.1007/BF01192134