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A note on Refracted Skew Brownian Motion with an application

Zaniar Ahmadi, Xiaowen Zhou

TL;DR

This work develops a perturbation-based framework for refracted skew Brownian motion (RSBM), a skew Brownian motion with a two-valued drift and local time at the skew level, to obtain its potential measures and transition densities via Laplace transform techniques. It provides explicit density formulas and asymptotic characterizations of the RSBM, including long-run stationary behavior and hitting probabilities, and shows consistency with prior SBM results while addressing specific two-drift configurations. The paper also offers practical sampling tools by proposing two quasi-random sampling schemes—mixtures of truncated normals and a deep neural network-based inverse-CDF model—to generate RSBM samples and to estimate risk metrics such as VaR and CVaR. These contributions yield both theoretical insights into RSBM dynamics and actionable methods for risk management and Monte Carlo simulation in asymmetric diffusion settings.

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.

A note on Refracted Skew Brownian Motion with an application

TL;DR

This work develops a perturbation-based framework for refracted skew Brownian motion (RSBM), a skew Brownian motion with a two-valued drift and local time at the skew level, to obtain its potential measures and transition densities via Laplace transform techniques. It provides explicit density formulas and asymptotic characterizations of the RSBM, including long-run stationary behavior and hitting probabilities, and shows consistency with prior SBM results while addressing specific two-drift configurations. The paper also offers practical sampling tools by proposing two quasi-random sampling schemes—mixtures of truncated normals and a deep neural network-based inverse-CDF model—to generate RSBM samples and to estimate risk metrics such as VaR and CVaR. These contributions yield both theoretical insights into RSBM dynamics and actionable methods for risk management and Monte Carlo simulation in asymmetric diffusion settings.

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.
Paper Structure (13 sections, 17 theorems, 109 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 17 theorems, 109 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Refracted Skew Brownian Motion and its exit problems
  3. Potential measures of RSBM
  4. Transition density of RSBM
  5. Long term behaviors
  6. Approximation and Simulation
  7. Using Truncated Normal Distribution
  8. Using Deep Neural Network
  9. Simulation Results
  10. Application in Risk Measurement
  11. Proof of Corollary \ref{['corr:sbm_one_drift']}
  12. Proof of Proposition \ref{['prop:sbm_one_drift']}
  13. Remarks on the joint density in Gairat and Shcherbakov gairat2017density

Key Result

Theorem 1

For $x,y,z \in \mathds{R}$ and $y \leq x < z$, we have and where

Figures (3)

  • Figure 1: Histogram and CDF of generated random samples for different models with deep neural network against the exact PDF and CDF of the model
  • Figure 2: Histogram and CDF of generated random samples for different models with truncated Normal approximation against the exact PDF and CDF of the model
  • Figure 3: Evaluating approximated VaR and CVaR using mixed truncated normal distribution and Monte-Carlo (MC) simulation methods for values greater than $\alpha = 0.2569$.

Theorems & Definitions (30)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • ...and 20 more