AI 中文总结
研究二维退化朗之万型随机偏微分方程数值逼近，提出半隐式米尔斯坦有限差分格式，经傅里叶分析得出稳定性条件与收敛率，嵌入多层蒙特卡罗框架降低计算成本，数值实验验证方法有效性及低复杂度。
AI 中文摘要
在这项工作中，我们研究二维退化朗之万型随机偏微分方程（SPDEs）的数值逼近。这些方程出现在随机动力学和数学金融等领域。为处理方程的混合确定性 - 随机结构及微分算子的退化性，我们提出半隐式米尔斯坦有限差分格式求解。通过傅里叶分析均方稳定性和收敛性，得出格式稳定的系数显式条件及收敛率。将其嵌入多层蒙特卡罗框架降低计算成本并推导理论复杂度。数值实验证实理论收敛率，表明多层蒙特卡罗策略以低计算成本达到与标准蒙特卡罗相当的精度，将复杂度从\(\mathcal{O}(\varepsilon^{-5})\)降至\(\mathcal{O}(\varepsilon^{-3})\)。结果表明半隐式米尔斯坦格式与多层蒙特卡罗技术结合为朗之万型SPDEs数值模拟提供有效方法。
英文摘要
In this work, we investigate the numerical approximation of degenerate Langevin-type stochastic partial differential equations (SPDEs) in two spatial dimensions. These SPDEs arise in stochastic dynamics and mathematical finance, among other applications. In order to handle the mixed deterministic-stochastic structure of the equation and the degeneracy of the differential operator, we propose a semi-implicit Milstein finite difference scheme for the numerical solution. Through the Fourier analysis of the mean-square stability and convergence, we derive explicit conditions on the coefficients under which the scheme is stable, jointly with explicit convergence rates in terms of the discretization parameters. We further embed the proposed scheme within a Multilevel Monte Carlo (MLMC) framework to reduce the computational cost associated with SPDE simulations, and we derive its theoretical computational complexity. Numerical experiments confirm theoretical convergence rates and show that the MLMC strategy achieves an accuracy comparable to standard Monte Carlo at a fraction of the computational cost, reducing the complexity from $\mathcal{O}(\varepsilon^{-5})$ to $\mathcal{O}(\varepsilon^{-3})$ for a target root-mean-square error $\varepsilon$. These results show that combining semi-implicit Milstein schemes with MLMC techniques provides an effective approach for the numerical simulation of Langevin-type SPDEs.
