Flots et series de Taylor stochastiques

SummaryWe 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.

[1]  Hiroshi Kunita,et al.  On the representation of solutions of stochastic differential equations , 1980 .

[2]  E. Platen.  A taylor formula for semimartingales solving a stochastic equation , 1981 .

[3]  Hiroshi Kunita,et al.  On the decomposition of solutions of stochastic differential equations , 1981 .

[4]  R. Azencott.  Densité des diffusions en temps petit: développements asymptotiques , 1984 .

[5]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[6]  P. Malliavin,et al.  Géométrie différentielle stochastique , 1978 .

[7]  Arthur J. Krener,et al.  The complexity of stochastic differential equations , 1981 .

[8]  T. Nagano.  Linear differential systems with singularities and an application to transitive Lie algebras , 1966 .

[9]  P. Meyer,et al.  Séminaire de probabilités X, Université de Strasbourg , 1971 .

[10]  B. Gaveau.  Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents , 1977 .

[11]  Robert Azencott,et al.  Formule de Taylor stochastique et developpement asymptotique d’integrales de Feynmann , 1982 .

[12]  H. Sussmann.  On the Gap Between Deterministic and Stochastic Ordinary Differential Equations , 1978 .

[13]  Hani J. Doss,et al.  Liens entre equations di erentielles stochastiques et ordinaires , 1977 .

[14]  M. Fliess,et al.  Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T. Chen , 1982 .

[15]  R. Palais.  A Global Formulation of the Lie Theory of Transformation Groups , 1957 .

[16]  Yuiti Yamato,et al.  Stochastic differential equations and Nilpotent Lie algebras , 1979 .