Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909772982255616 |
|---|---|
| author | Dang, Thanh Zhu, Lingjiong |
| author_facet | Dang, Thanh Zhu, Lingjiong |
| contents | Via a Bismut-Elworthy-Li formula from [KPP23], we derive uniform gradient estimates for transition semigroups associated with stochastic differential equations driven by a large class of cylindrical Lévy processes which includes the important case of cylindrical $α$-stable processes. As the first application, we formulate a Stein's method for quantitative approximation of the invariant measure of these stochastic differential equations in Wasserstein distance. As the second and main application, we study Euler-Maruyama numerical schemes of stochastic differential equations driven by stable Lévy processes with i.i.d. stable components and obtain a uniform-in-time approximation error in Wasserstein distance. Our approximation error has a linear dependence on the stepsize, which is expected to be tight, as can be seen from an explicit calculation for the case of an Ornstein-Uhlenbeck process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12502 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes Dang, Thanh Zhu, Lingjiong Probability Via a Bismut-Elworthy-Li formula from [KPP23], we derive uniform gradient estimates for transition semigroups associated with stochastic differential equations driven by a large class of cylindrical Lévy processes which includes the important case of cylindrical $α$-stable processes. As the first application, we formulate a Stein's method for quantitative approximation of the invariant measure of these stochastic differential equations in Wasserstein distance. As the second and main application, we study Euler-Maruyama numerical schemes of stochastic differential equations driven by stable Lévy processes with i.i.d. stable components and obtain a uniform-in-time approximation error in Wasserstein distance. Our approximation error has a linear dependence on the stepsize, which is expected to be tight, as can be seen from an explicit calculation for the case of an Ornstein-Uhlenbeck process. |
| title | Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes |
| topic | Probability |
| url | https://arxiv.org/abs/2402.12502 |