基于数值平滑的多层蒙特卡洛方法用于稳健高效地计算概率与密度
Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities
AI总结：
本文将数值平滑技术扩展到多层蒙特卡洛方法中，通过求根与一维数值积分相结合，显著改善了低正则性泛函下的强收敛性、复杂度与稳健性，并成功应用于概率计算、不连续收益期权定价及密度估计。
摘要：
多层蒙特卡洛（MLMC）方法在估计随机微分方程（SDE）解的泛函期望方面非常高效。然而，当泛函的正则性较低时，MLMC估计量可能不稳定，并且具有较差的（非规范的）复杂度。为了克服这一问题，我们将此前在（Quantitative Finance, 23(2), 209-227, 2023）中针对确定性求积方法提出的数值平滑思想扩展到MLMC框架中。该数值平滑技术基于求根方法，并结合对单个精心选取变量的一维数值积分。本研究的动机包括事件概率的计算、具有不连续收益的期权定价，以及在必须对底层随机过程进行离散化的动力学场景下的密度估计问题。分析与数值实验表明，数值平滑显著改善了强收敛性，从而提升了MLMC方法的复杂度和稳健性（通过使深层峰度保持有界）。特别地，我们证明，在使用Euler–Maruyama格式时，数值平滑能够凭借最优方差衰减率恢复Lipschitz泛函所获得的MLMC复杂度。对于Milstein格式，即使针对上述非光滑被积函数，数值平滑也能恢复规范的MLMC复杂度。最后，我们的方法能够高效地估计一元和多元密度函数。
英文摘要：
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.
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