First exit and Dirichlet problem for the nonisotropic tempered $\alpha$-stable processes

Abstract

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $\alpha$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{{\tau }_D}\right|$ and the first exit time ${\tau }_D$ are firstly obtained. It is found that the probability density function of $\left|X_{{\tau }_D}\right|$ or ${\tau }_D$ exponentially decays with the increase of $\left|X_{{\tau }_D}\right|$ or ${\tau }_D$, and $\mathrm{E}\left[{\tau }_D\right]\sim \left|\mathrm{E}\left[X_{{\tau }_D}\right]\right|$,\ $\mathrm{E}\left[{\tau }_D\right]\sim \mathrm{E}\left[{\left|X_{{\tau }_D}-\mathrm{E}\left[X_{{\tau }_D}\right]\right|}^2\right]$. Since ${\mathrm{\Delta }}_m^{\alpha /2,\lambda }$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator ${\mathrm{\Delta }}_m^{\alpha /2,\lambda }$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.

23 pages, 5 figures