  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.
