On the existence of a time inhomogeneous skew Brownian motion and some related laws

TL;DR

This paper constructs a solution for a time inhomogeneous skew Brownian motion, explores related laws, and computes joint distributions involving the process, local time, and straddling time, extending foundational work from 1983.

Contribution

It provides a new construction for the inhomogeneous skew Brownian motion and derives explicit joint laws involving key process features.

Findings

Explicit joint law of the process, local time, and straddling time

Construction of solutions for the inhomogeneous skew Brownian motion

Extension of Weinryb's 1983 foundational results

Abstract

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation, which first appear in a seminal paper by Sophie Weinryb (1983). We investigate some laws related to the constructed process. In particular, using the description of the straddling excursion above a deterministic time, we compute the joint law of the process, its local time and its straddling time.