First exit and Dirichlet problem for the nonisotropic tempered α-stable processes

2019/01/08 by Liu, Xing, Deng, Weihua
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1901.03204

Abstract

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_τD| and the first exit time τ_D are firstly obtained. It is found that the probability density function of |X_τD| or τ_D exponentially decays with the increase of |X_τD| or τ_D, and E[τ_D]∼ |E[X_τD]|, E[τ_D]\simE[|X_τD-E[X_τD]|²] . Since \mathrmΔ^α/2,λ_m is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator \mathrmΔ^α/2,λ_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.

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