Asymptotics of the hitting probability for a small sphere and a two dimensional Brownian motion with discontinuous anisotropic drift

Abstract

We provide an approximation of the hitting probability for a small sphere for the following two dimensional process: In x-direction it is just a Brownian motion with positive constant drift, whereas in y-direction the process ${\mathit{Y}}_{\mathit{t}}$ is a Brownian motion with drift given by a negative constant times the sign of ${\mathit{Y}}_{\mathit{t}}$. This process can be seen as the solution of a certain stochastic optimal control problem. It turns out that the approximating function can be expressed as the sum of a term involving a modified Bessel function and an ordinary Lebesgue integral.