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

TL;DR

This work proves that fully discrete schemes for SPDEs driven by Lévy noise can achieve $L^p$-strong convergence orders that are effectively independent of the moment order $p$ (for all $p\ge2$) and nearly $1/2$ in both space and time. It analyzes two noise regimes: finite-activity Lévy noise via a jump-adapted time grid, and infinite-activity Lévy noise through Le's quantitative John--Nirenberg inequality, delivering $p$-uniform error bounds of the form $C_{p,T}(N^{-(1-\delta)/2}+\Delta t^{(1-\delta)/2})$. The scheme uses spectral Galerkin spatial discretization and Euler-type time stepping; it carefully handles Poisson integrals, stochastic convolutions, and nested conditional norms to overcome the obstructions posed by temporal Hölder continuity. The results extend numerical analysis for SPDEs with Lévy noise, providing robust convergence guarantees across jump structures and improving reliability for simulations of jump-dominated stochastic systems.

Abstract

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.

Theorems & Definitions (18)

• Remark 2.1
• Lemma 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• proof : Proof of Theorem \ref{['the:convergenceorder-jumpadapted']}
• Lemma 4.1
• Lemma 4.2
• Proposition 4.3
• Proposition 4.4