$L^p$-strong convergence orders of fully discrete schemes for the SPDE driven by Lévy noise
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arXiv
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| Format: | Preprint |
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2024
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| author | Chen, Chuchu Dang, Tonghe Hong, Jialin Lei, Ziyi |
| author_facet | Chen, Chuchu Dang, Tonghe Hong, Jialin Lei, Ziyi |
| contents | It is well known that for a stochastic differential equation driven by Lévy noise, the temporal Hölder continuity in $L^p$ sense of the exact solution does not exceed $1/p$. This leads to that the $L^p$-strong convergence order of a numerical scheme will vanish as $p$ increases to infinity if the temporal Hölder continuity of the solution process is directly used. A natural question arises: can one obtain the $L^p$-strong convergence order that does not depend on $p$? In this paper, we provide a positive answer for fully discrete schemes of the stochastic partial differential equation (SPDE) driven by Lévy noise. Two cases are considered: the first is the linear multiplicative Poisson noise with $ν(χ)<\infty$ and the second is the additive Poisson noise with $ν(χ)\leq\infty$, where $ν$ is the Lévy measure and $χ$ is the mark set. For the first case, we present a strategy by employing the jump-adapted time discretization, while for the second case, we introduce the approach based on the recently obtained Lê's quantitative John--Nirenberg inequality. We show that proposed schemes converge in $L^p$ sense with orders almost $1/2$ in both space and time for all $p\ge2$, which contributes novel results in the numerical analysis of the SPDE driven by Lévy noise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-strong convergence orders of fully discrete schemes for the SPDE driven by Lévy noise Chen, Chuchu Dang, Tonghe Hong, Jialin Lei, Ziyi Numerical Analysis Probability It is well known that for a stochastic differential equation driven by Lévy noise, the temporal Hölder continuity in $L^p$ sense of the exact solution does not exceed $1/p$. This leads to that the $L^p$-strong convergence order of a numerical scheme will vanish as $p$ increases to infinity if the temporal Hölder continuity of the solution process is directly used. A natural question arises: can one obtain the $L^p$-strong convergence order that does not depend on $p$? In this paper, we provide a positive answer for fully discrete schemes of the stochastic partial differential equation (SPDE) driven by Lévy noise. Two cases are considered: the first is the linear multiplicative Poisson noise with $ν(χ)<\infty$ and the second is the additive Poisson noise with $ν(χ)\leq\infty$, where $ν$ is the Lévy measure and $χ$ is the mark set. For the first case, we present a strategy by employing the jump-adapted time discretization, while for the second case, we introduce the approach based on the recently obtained Lê's quantitative John--Nirenberg inequality. We show that proposed schemes converge in $L^p$ sense with orders almost $1/2$ in both space and time for all $p\ge2$, which contributes novel results in the numerical analysis of the SPDE driven by Lévy noise. |
| title | $L^p$-strong convergence orders of fully discrete schemes for the SPDE driven by Lévy noise |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2412.05539 |