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 α -stable process X_t . The upper bounds of all moments of the first exit position | X_τ _D| and the first exit time τ _D are explicitly obtained. It is found that the probability density function of | X_τ _D| or τ _D exponentially decays with the increase of | X_τ _D| or τ _D , and E[ τ _D] ∼E[ | X_τ _D-E[ X_τ _D] | ^2] , E[ τ _D] ∼| E[ X_τ _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.