In this paper, we investigate the properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by the infinite-dimensional Wiener process and Poisson random measure with the Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which is determined by two parameters, i.e grid density $n \in \mathbb{N}_{+}$ and truncation dimension parameter $M \in \mathbb{N}_{+},$ is of the order $n^{-1/2}+\delta(M)$ such that $\delta(\cdot)$ is positive and decreasing to $0$. The paper introduces the complexity model and provides proof for the upper complexity bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both $n$ and $M.$ The complexity is measured in terms of upper bound for mean-squared error and compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation are also reported.

翻译：本文研究了标准与多层蒙特卡洛方法在无限维维纳过程与泊松随机测度驱动的随机微分方程弱逼近中的性质，其中支付函数满足利普希茨条件。截断维度随机化数值方案的误差由两个参数（网格密度$n \in \mathbb{N}_{+}$与截断维度参数$M \in \mathbb{N}_{+}$）决定，其阶数为$n^{-1/2}+\delta(M)$，其中$\delta(\cdot)$为正且递减至零。本文引入了复杂度模型，并证明了多层蒙特卡洛方法的上界复杂度，该复杂度取决于$n$和$M$的两个递增参数序列。复杂度通过均方误差的上界衡量，并与标准蒙特卡洛算法的复杂度进行了比较。此外，本文报告了数值实验的结果以及Python和CUDA C实现。