Stochastic Volterra Equations for the Local Times of Spectrally Positive Stable Processes

Wei Xu

Wei Xu

Jan 2024

0被引用

0笔记

摘要原文

This paper is concerned with the evolution dynamics of local times of a spectrally positive stable process in the spatial direction. The main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the L\'evy measure. This helps us to provide not only a simple proof for the H\"older regularity, but also a uniform upper bound for all moments of the H\"older coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.