First exit and Dirichlet problem for the nonisotropic tempered α-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 |X-tau D| and the first exit time tau(D) are explicitly obtained. It is found that the probability density function of |X-tau D| or tau(D) exponentially decays with the increase of |X-tau D| or tau(D), and E[tau(D)]similar to E[|X tau(D)-E[X tau(D)]|2], E[tau(D)]similar to|E[X tau(D)]|. 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.