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

TL;DR

This paper studies the local times of spectrally positive stable processes, showing they satisfy a stochastic Volterra equation driven by a Poisson measure, leading to new regularity results and exponential-affine representations.

Contribution

It introduces a stochastic Volterra equation characterization for local times of stable processes, enabling new proofs of regularity and Laplace functional representations.

Findings

Local times satisfy a stochastic Volterra equation driven by Poisson measure

Established H"older regularity and uniform moment bounds for local times

Derived exponential-affine Laplace functional representation

Abstract

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.