标题： $L^p$-强收敛阶的全离散格式用于由Lévy噪声驱动的SPDE 显示英文标题

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

摘要： 众所周知，对于由Lévy噪声驱动的随机微分方程，精确解在$L^p$意义下的时间Hölder连续性不会超过$1/p$。 这导致如果直接使用解过程的时间Hölder连续性，当$p$增加到无穷大时，数值方案的$L^p$-强收敛阶将消失。 一个自然的问题出现：能否获得不依赖于$p$的$L^p$-强收敛阶？ 在本文中，我们为由Lévy噪声驱动的随机偏微分方程（SPDE）的全离散格式提供了肯定的答案。 考虑了两种情况：第一种是线性乘法泊松噪声，其中$\nu(\chi)<\infty$，第二种是加法泊松噪声，其中$\nu(\chi)\leq\infty$，其中$\nu$是 Lévy 测度，$\chi$是标记集。 对于第一种情况，我们通过采用跳跃适应的时间离散化提出了一种策略，而对于第二种情况，我们引入了基于最近获得的 Lê 的定量 John--Nirenberg 不等式的办法。 我们证明了所提出的格式在所有$p\ge2$的空间和时间中几乎以$1/2$的阶数在$L^p$意义下收敛，这为由 Lévy 噪声驱动的 SPDE 的数值分析提供了新的结果。 显示英文摘要

摘要： 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.