论文标题
具有数值平滑的多级蒙特卡洛，以稳健有效地计算概率和密度
Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities
论文作者
论文摘要
多级蒙特卡洛（MLMC）方法对于估算对随机微分方程（SDE）的解决方案功能的预期高度有效。但是，在功能较低的情况下，MLMC估计器可能不稳定，并且具有较差的（非规范）复杂性。为了克服这个问题，我们将先前介绍的数值平滑概念扩展为（定量金融，23（2），209-227，2023），在确定性正交方法的背景下，将其扩展到MLMC设置。数值平滑技术基于与单一数值相对于单个精心挑选的变量的一维数值集成的根发现方法。这项研究是由事件概率，不连续回报的定价选项的计算以及动态的密度估计问题的计算，在这种情况下，基本随机过程的离散化是必要的。分析和数值实验表明，数值平滑显着改善了强大的收敛性，因此，MLMC方法的复杂性和鲁棒性（通过在深层界定的峰度中）。特别是，我们表明，由于使用Euler-Maruyama方案时，由于最佳方差衰减率，数值平滑能够恢复为Lipschitz功能获得的MLMC复杂性。对于米尔斯坦方案，即使对于上面提到的非平滑整合体，数值平滑也可以恢复规范的MLMC复杂性。最后，我们的方法有效估计单变量和多变量密度函数。
The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.
