Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an F-semi-martingale M and a finite cubic variation process 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 Ito and Ito-Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when is a Holder continuous process and sigma is only Holder in space. Using an Ito formula for reversible semimartingales, we also show existence of a solution when is a Brownian motion and or is only continuous.