Stochastic Volterra Equations for Local Times of Spectrally Positive L\'evy Processes with Gaussian Components

TL;DR

This paper studies the evolution of local times of spectrally positive Lévy processes with Gaussian components, using stochastic Volterra equations driven by Gaussian white noise and Poisson measures, establishing existence, uniqueness, and properties.

Contribution

It introduces a stochastic Volterra equation framework for local times of Lévy processes with Gaussian components, proving strong solutions and deriving new properties and representations.

Findings

Established strong existence and uniqueness of solutions.

Derived a comparison principle and stochastic flow for local times.

Provided an exponential-affine Laplace functional representation.

Abstract

Following our previous work [68], this paper continues to investigate the evolution dynamics of local times of spectrally positive L\'evy processes with Gaussian components in the spatial direction. We prove that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, local times at positive line are equal in law to the unique solution of a stochastic Volterra equation driven by a Gaussian white noise and two Poisson random measures with convolution kernel given in terms of the scale function. Also, we obtain several equivalent stochastic equations by using the potential theoretic techniques and prove the strong existence and uniqueness by using the generalized Yamada-Watanabe theorems. Armed with the stochastic Volterra representation, we then establish a comparison principle for the local times of spectrally positive L\'evy processes with various drifts or stopped when local times at zero exceed different given values, which proposes a stochastic flow enjoying the branching property. And also, we explore some novel properties of local times in the spatial direction including uniform moment estimates, $(1/2-\varepsilon)$-H\"older continuity and maximal inequality. By using the method of duality, we provide an exponential-affine representation of the Laplace functional in terms of the unique non-negative solution of a path-dependent nonlinear Volterra equation associated with the Laplace exponent of L\'evy process. This gives another perspective on the evolution dynamics of local times in the spatial direction.