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 $\xi$ 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 It\^{o} and It\^{o}--Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when $\xi$ is a H\"{o}lder continuous process and $\sigma$ is only H\"{o}lder in space. Using an It\^{o} formula for reversible semimartingales, we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.