Résumé: In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by a $\mathbb F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is strongly predictable in some technical sense: that condition implies in particular that $R$ is weak Dirichlet, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the "diffusion" coefficient multiplying $\xi$ to 1. We use generalized Itô and Itô-Wentzell type formulae. A similar method allows to discuss existence and uniqueness theorem when $\xi$ is a Hölder continuous process and $\sigma$ is only Hölder in space. Using an Itô formula for reversible semimartingales we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.

Code(s) de Classification MSC2000:  60H05 Stochastic integrals ;  60H10 Stochastic ordinary differential equations [See also 34F05] ;  60G18 Self-similar processes ;  60G20 Generalized stochastic processes ; 

Mots Clés: Finite cubic variation; Itô-Wentzell formula; Stochastic differential equation; Hölder processes; weak Dirichlet processes.

Langue du texte: Anglais

Article reçu: 2005-04-27