A note on Refracted Skew Brownian Motion with an application

For refracted skew Brownian motion (skew Brownian motion with two-valued drift), adopting a perturbation approach we find expressions of its potential densities. As applications, we recover its transition density and study its long-time asymptotic behaviors. In addition, we also compare with previous results on transition densities for skew Brownian motions. We propose two approaches for generating quasi-random samples by approximating the cumulative distribution function and discuss their risk measurement application.