$L^p$-strong convergence orders of fully discrete schemes for the SPDE driven by L\'evy noise

TL;DR

This paper establishes that fully discrete numerical schemes for SPDEs driven by Lévy noise achieve an $L^p$-strong convergence order close to 1/2 in both space and time, independent of $p$, overcoming previous limitations related to Hölder continuity.

Contribution

The paper introduces new strategies for analyzing convergence orders of numerical schemes for SPDEs with Lévy noise, achieving $p$-independent convergence rates near 1/2.

Findings

Convergence order close to 1/2 in space and time for all $p\,\geq 2$.

Strategies employing jump-adapted discretization and John--Nirenberg inequality.

Results contribute novel insights into numerical analysis of Lévy-driven SPDEs.

Abstract

It is well known that for a stochastic differential equation driven by L\'evy noise, the temporal H\"older 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 H\"older 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 L\'evy noise. Two cases are considered: the first is the linear multiplicative Poisson noise with $\nu(\chi)<\infty$ and the second is the additive Poisson noise with $\nu(\chi)\leq\infty$, where $\nu$ is the L\'evy measure and $\chi$ 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 L\^e'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 L\'evy noise.