Skew Brownian Motion with Two-Valued Drift and Applications in Optimal Control

Speaker: Xiaowen Zhou
Affiliation: Concordia University, Canada

Abstract

Motivated by its applications in the optimal dividend problem in actuarial science, we consider a skew Brownian motion with two-valued drift as the unique solution to stochastic differential equation

dXt = μ−1{Xt<a} + μ+1{Xt>a} dt + dBt + βdL a t (X)

driven by Brownian motion B and symmetric local time process L a (X) at level a with drift coefficients μ− and μ+ and skewness −1 < β < 1. Such a process can be identified as a toy model for spatial regime switching.

In this talk we first apply the Ito-Tanaka-Meyer formula together with a martingale approach to find Laplace transforms of exit times for the skew Brownian motion.

We further consider an optimal control problem in which we look for an optimal dividend strategy that maximizes the expected accumulated present value of dividends until ruin for the skew Brownian surplus process. By showing a verification theorem on the associated Hamilton–Jacobi–Bellman inequalities, we identify conditions for different barrier strategies to be optimal and observe that certain band strategies involving two dividend barriers can be optimal. We also illustrate how the optimal strategies are affected by different choices of drifts and skewness.

This talk is based on joint work with Zhongqin Gao.