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：我们针对前期工作(Ben Hammouda et al., 2023)提出了一种新型替代方案，以提升随机反应网络(SRNs)中稀有事件概率的蒙特卡洛(MC)估计效率。遵循(Ben Hammouda et al., 2023)的核心理念，通过建立概率测度类中最优重要性采样(IS)参数确定与随机最优控制公式之间的关联（对应求解方差最小化问题），推导出高效的路径依赖测度变换。本研究提出创新方法解决维度灾难问题：运用马尔可夫投影(MP)思想将问题映射到显著低维空间。该模型降阶技术的输出是保留原始高维SRN系统边际分布的低维SRN（甚至可为一维系统）。通过离散$L^2$回归求解相关优化问题，获得投影过程的动力学方程。通过求解降维SRN对应的投影哈密顿-雅可比-贝尔曼(HJB)方程，获得投影IS参数，再将其映射回原始全维SRN系统，最终构建出全维SRN稀有事件概率的高效IS-MC估计器。理论分析与数值实验表明，所提出的MP-HJB-IS方法能显著降低MC估计器方差，在稀有事件区域实现比标准MC估计器更低的计算复杂度。