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Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components

Wei Xu

TL;DR

This work develops a stochastic Volterra framework for local times of spectrally positive Lévy processes with Gaussian components, conditioned on finite first hitting time of zero, by representing the spatial evolution $L^\xi_ zeta$ as the unique nonnegative solution to an SVE driven by Gaussian white noise and Poisson measures with scale-function kernels. The authors establish a robust convergence from compound Poisson approximations, prove a comparison principle for L under drift variations, and obtain moment and Hölder regularity properties, as well as an affine Laplace representation via a path-dependent nonlinear Volterra equation. Central to the approach are scale-function identities, martingale-measure stochastic calculus, and a careful asymptotic analysis that yields strong existence and uniqueness under a finiteness condition on the Lévy tail, along with a clear pathwise description of the stochastic flow of local times. The results deepen the understanding of local-time dynamics in the spatial direction and have potential implications for connections to random trees, branching structures, and excursion theory in Lévy environments.

Abstract

Following our previous work [68], this paper continues to investigate the evolution dynamics of local times of spectrally positive Lévy 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 Lévy 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, $(1/2-\varepsilon)$-Hölder 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 Lévy process. This gives another perspective on the evolution dynamics of local times in the spatial direction.

Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components

TL;DR

This work develops a stochastic Volterra framework for local times of spectrally positive Lévy processes with Gaussian components, conditioned on finite first hitting time of zero, by representing the spatial evolution as the unique nonnegative solution to an SVE driven by Gaussian white noise and Poisson measures with scale-function kernels. The authors establish a robust convergence from compound Poisson approximations, prove a comparison principle for L under drift variations, and obtain moment and Hölder regularity properties, as well as an affine Laplace representation via a path-dependent nonlinear Volterra equation. Central to the approach are scale-function identities, martingale-measure stochastic calculus, and a careful asymptotic analysis that yields strong existence and uniqueness under a finiteness condition on the Lévy tail, along with a clear pathwise description of the stochastic flow of local times. The results deepen the understanding of local-time dynamics in the spatial direction and have potential implications for connections to random trees, branching structures, and excursion theory in Lévy environments.

Abstract

Following our previous work [68], this paper continues to investigate the evolution dynamics of local times of spectrally positive Lévy 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 Lévy 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, -Hölder 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 Lévy process. This gives another perspective on the evolution dynamics of local times in the spatial direction.

Paper Structure

This paper contains 33 sections, 72 theorems, 512 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Preliminaries
  4. Scale functions
  5. Stochastic integrals driven by martingale measures
  6. Local times of compound Poisson processes
  7. Compound Poisson processes
  8. Stochastic Volterra representation of $L^{Y}_{k}$
  9. Poisson approximation
  10. Stochastic Volterra representation of $L^\xi_\zeta$
  11. Asymptotic assumptions
  12. Auxiliary lemmas
  13. Asymptotics of rescaled resolvent
  14. Uniform moment estimates
  15. $C$-tightness
  16. ...and 18 more sections

Key Result

Theorem 1.1

For each $\zeta\geq 0$, the process $L^\xi_\zeta$ equals in distribution to the unique non-negative continuous solution of the stochastic Volterra equation where $N_0(dz, dy)$ is a Poisson random measure on $(0,\infty)^2$ with intensity $\bar{\nu}(y)\,dz\,dy$, $B_c(ds,dz)$ is a Gaussian white noise on $(0,\infty)^2$ with intensity $2c\cdot ds\, dz$ and $\widetilde{N}_\nu(ds,dz,dy)$ is a compensat

Figures (1)

  • Figure 1.1: A sample path of spectrally positive Lévy processes containing three typical kinds of excursions away from level $x$: (i) $\mathcal{E}_+^x$ excursions that are completely above $x$ (the green trajectories); (ii) $\mathcal{E}_-^x$ excursions that are completely blow $x$ (the blue trajectory); (iii) $\mathcal{E}_\pm^x$ excursions that move below $x$ until jumping into $(x,\infty)$ and then stay above $x$ up to hitting $x$ (the magenta trajectory).

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 68 more