Stochastic differential equations driven by relative martingales

TL;DR

This paper develops a general framework for relative martingales driven by stochastic differential equations, extending their definition and applying the results to construct solutions for skew Brownian motion.

Contribution

It introduces a broader class of relative martingales by relaxing the null condition on a set and explores their structural properties and applications in stochastic differential equations.

Findings

Extended the notion of relative martingales to non-null processes.

Provided structural properties of the new class of relative martingales.

Constructed solutions for skew Brownian motion using the new framework.

Abstract

This paper contributes to the study of relative martingales. Specifically, for a closed random set $H$, they are processes null on $H$ which decompose as $M=m+v$, where $m$ is a c\`adl\`ag uniformly integrable martingale and, $v$ is a continuous process with integrable variations such that $v_{0}=0$ and $dv$ is carried by $H$. First, we extend this notion to stochastic processes not necessarily null on $H$, where $m$ is considered local martingale instead of a uniformly integrable martingale. Thus, we provide a general framework for the new larger class of relative martingales by presenting some structural properties. Second, as applications, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the above mentioned new class. In addition, we investigate stochastic differential equations driven by a relative martingale.