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

## TL;DR

This paper investigates the first exit and Dirichlet problems for nonisotropic tempered alpha-stable processes, providing bounds, decay properties, and a Feynman-Kac representation verified through numerical experiments.

## Contribution

It introduces bounds on moments, decay properties, and a Feynman-Kac formula for nonisotropic tempered alpha-stable processes, advancing understanding of their exit behaviors and solutions to related PDEs.

## Key findings

- Moments of exit position and time are bounded and decay exponentially.
- Expected exit time relates linearly to the expected exit position.
- Numerical experiments verify the Feynman-Kac representation for the Dirichlet problem.

## 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 $\mathrm{\Delta}^{\alpha/2,\lambda}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrm{\Delta}^{\alpha/2,\lambda}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.