We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of (Ben Hammouda et al., 2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete $L^2$ regression. By solving the resulting projected Hamilton-Jacobi-Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

翻译：本文针对我们此前工作（Ben Hammouda等，2023）提出了一种全新的替代方案，旨在提升随机反应网络（SRNs）中稀有事件概率的蒙特卡洛（MC）估计效率。延续（Ben Hammouda等，2023）的核心思路，通过建立一类概率测度内最优重要性采样（IS）参数求解与随机最优控制公式之间的关联，我们推导出高效的路径依赖测度变换，该变换对应于解决方差最小化问题。本研究提出了一种应对维数灾难的新方法：通过马尔可夫投影（MP）思想将问题映射至显著低维空间。该模型降阶技术的输出是低维SRN（甚至可能降至一维），且能保持原始高维SRN系统的边际分布。投影过程的动力学通过离散$L^2$回归求解相关优化问题获得。通过求解降维SRN对应的投影Hamilton-Jacobi-Bellman（HJB）方程，我们获得投影IS参数，继而将其映射回原始全维SRN系统，由此构建全维SRN稀有事件概率的高效IS-MC估计量。理论分析与数值实验表明，所提出的MP-HJB-IS方法能大幅降低MC估计量方差，相较于标准MC估计量在稀有事件区间内具有更低的计算复杂度。