Resolution of the skew Brownian motion equations with stochastic calculus for signed measures

TL;DR

This paper advances stochastic calculus for signed measures, characterizes related martingales and Brownian motions, and constructs solutions for homogeneous and inhomogeneous skew Brownian motion equations using these new tools.

Contribution

It introduces new results in stochastic calculus for signed measures and applies them to solve skew Brownian motion equations.

Findings

Characterization of martingales under signed measures

Representation of uniformly integrable martingales as relative martingales

Construction of solutions for skew Brownian motion equations

Abstract

Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class $\Sigma(H)$ which appeared for the first time in \cite{f} and studied in \cite{f,e,o}. The second part is devoted to the construction of solutions for the \textbf{homogeneous skew Brownian motion equation} and for the \textbf{inhomogeneous skew Brownian motion equation}. To do this, our ingredients are the techniques and results developed in the first part that we apply on some stochastic processes borrowed from the theory of stochastic calculus for signed measures. Our methods are inspired by those used by Bouhadou and Ouknine in \cite{siam}. Moreover, their solution of the inhomogeneous skew Brownian motion equation is a particular case of those we propose in this paper.