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

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 explicitly 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 \(\mathrm{E}\left[ \tau _D\right] \sim \mathrm{E}\left[ \left| X_{\tau _D}-\mathrm{E}\left[ X_{\tau _D}\right] \right| ^2\right]\), \(\mathrm{E}\left[ \tau _D\right] \sim \left| \mathrm{E}\left[ X_{\tau _D}\right] \right|\). Next, we obtain the Feynman–Kac representation of the Dirichlet problem by employing the semigroup theory. Furthermore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.