A note on Refracted Skew Brownian Motion with an application

TL;DR

This paper analyzes refracted skew Brownian motion, deriving its potential densities, transition densities, and asymptotic behaviors, and introduces methods for quasi-random sampling with applications in risk measurement.

Contribution

It provides new explicit expressions for potential and transition densities of refracted skew Brownian motion and proposes novel sampling methods for risk assessment.

Findings

Derived explicit potential densities using perturbation methods.

Recovered transition densities and analyzed long-term behavior.

Proposed two quasi-random sampling approaches for risk measurement.

Abstract

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.