Brownian motion with two-valued drift and variance
数学专题报告
报告题目(Title)：Brownian motion with two-valued drift and variance
报告人(Speaker)：周晓文教授 (加拿大 Concordia大学)
地点(Place)：后主楼 12 层 1220 报告厅
时间(Time)：2024 年 5 月 23 日(星期四)上午 8:00-9:00
邀请人(Inviter)：李增沪
报告摘要
Motivation by problems in stochastic control, we consider the unique solution $X$ to the following SDE
\dd X_t=(\mu_1 \mathbf{1}_{\{X_t \le 0\}}+\mu_2 \mathbf{1}_{\{X_t> 0\}})\dd t+(\sigma_1\mathbf{1}_{\{X_t\le 0\}}+\sigma_2\mathbf{1}_{\{X_t> 0\}})\dd B_t
for $\mu_1, \mu_2\in\mathbb{R}$ and $\sigma_1, \sigma_2>0$.
For $\mu_1=\mu_2$ an explicit expression for transition density of $X$ was obtained by Keilson and Wellner (1978). For $\sigma_1=\sigma_2$ the transition density was obtained by Karatzas and Shreve (1984). But the transition density for general $X$ was not known.
We first solve the exit problem to process $X$, and then adopt a perturbation approach to find an expression of potential measure for $X$. The transition density is found by inverting the Laplace transform.
This talk is based on joint work with Zengjing Chen and Panyu Wu.
