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

Wei Xu

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 Lévy measure. This helps us to provide not only a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the Hölder 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.

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

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 Lévy measure. This helps us to provide not only a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the Hölder 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.

Paper Structure

This paper contains 31 sections, 51 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Overview of main results
  3. Methodologies
  4. Related Literature
  5. Preliminaries and main results
  6. Spectrally positive stable processes
  7. Main results
  8. Local times of compound Poisson processes
  9. Branching representation
  10. Stochastic Volterra representation
  11. Stochastic Volterra equation for $L^{\xi}_\zeta$
  12. Compound Poisson approximation
  13. Proofs for Theorem \ref{['MainThm.01']}(1)-(2)
  14. Proofs for auxiliary lemmas
  15. Upper bounds for resolvents
  16. ...and 16 more sections

Key Result

Theorem 2.1

We have the following:

Theorems & Definitions (65)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9: Hölder continuity
  • Corollary 2.10: Maximal inequality
  • ...and 55 more