Motivated by problems in stochastic control, we consider the unique solution X to the following SDE

dX_t = (μ_1 \mathbf{1}{Xt≤0} + μ2 \mathbf{1}{Xt>0})dt + (σ1 \mathbf{1}{Xt≤0} + σ2 \mathbf{1}{Xt>0})dBt for μ_1, μ_2 ∈ R and σ_1, σ_2 > 0.

For μ_1 = μ_2 an explicit expression for transition density of X was obtained by Keilson and Wellner (1978). For σ_1 = σ_2 the transition density was obtained by Karatzas and Shreve (1984).

But the transition density for X was not known. To find the transition density, 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 density is obtained by inverting the Laplace transform. We will also present more recent work on multi-threshold Brownian motion.

This talk is based on joint work with Zengjing Chen, Panyu Wu and Weihai Zhang, and with Lina Ji and Chuyang Li.