Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910829868220416 |
|---|---|
| author | Xu, Wei |
| author_facet | Xu, Wei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components Xu, Wei Probability 2020 subject classifications: Primary 60G51, 60J55, 60H20, secondary 60G22, 60F17, 60G55 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. |
| title | Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components |
| topic | Probability 2020 subject classifications: Primary 60G51, 60J55, 60H20, secondary 60G22, 60F17, 60G55 |
| url | https://arxiv.org/abs/2411.15485 |