Explicit Transition Density Functions of Skew Brownian Motions with Two-Valued Drift

TL;DR

This paper derives explicit transition density functions for skew Brownian motion with two-valued drift, establishing its Markov properties and analyzing conditions for transience or recurrence.

Contribution

It provides the first explicit formulas for the transition densities of SBM with two-valued drift and characterizes its Markovian and recurrence properties.

Findings

Explicit transition density functions derived for all t>0.

SBM with two-valued drift is shown to be a strong Markov process.

Conditions for transience and recurrence are identified.

Abstract

In this article, we derive the explicit transition density functions of skew Brownian motion (SBM in abbreviation) with two-valued drift for all $t>0$. As an important step of this result, it is also shown in this paper that SBM with two-valued drift is a strong Markov process by finding its symmetrizing measure and canonical scale function, from which one can tell what values of the drift make such a process transient or recurrent.