Distance between two skew Brownian motions as a SDE with jumps and law of the hitting time

TL;DR

This paper models the distance between two skew Brownian motions with different initial points and skewness, revealing a jump SDE and deriving the distribution of their first hitting time's local time as a Beta distribution.

Contribution

It introduces a novel SDE with jumps for the distance between skew Brownian motions and characterizes the law of their hitting time, extending previous coalescence results.

Findings

Distance evolution described by a jump SDE

Distribution of local time at hitting time as Beta

Extension of coalescence law for different skewness

Abstract

In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burdzy and Chen (2001), where the law of coalescence of two skew Brownian motions with the same skewness coefficient is computed.