Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

Abstract

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb{F}$-semimartingale M and a finite cubic variation process ξ 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 ξ to 1. We use generalized Itô and Itô–Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when ξ is a Hölder continuous process and σ is only Hölder in space. Using an Itô formula for reversible semimartingales, we also show existence of a solution when ξ is a Brownian motion and σ is only continuous.