240118 Chiheb Hammouda

Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, and supply chain networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and introduce two novel importance sampling (IS) methods to enhance the Monte Carlo (MC) estimator's performance. The key challenge in the IS framework is choosing an appropriate change of probability measure to significantly reduce variance, and which often requires insights into the underlying problem. To address this, we propose a generic approach to obtain an efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control (SOC) formulation. When solving the resulting SOC problem, we tackle the curse of dimensionality in two ways: the first is a learning-based method that uses a neural network to approximate the value function and where the IS parameters are determined via a stochastic optimization algorithm. The second is a dimensionality reduction technique based on the Markovian Projection concept, where we map the problem to a significantly lower dimensional space, while preserving the marginal distribution of the original SRN system. We then solve a Hamilton-Jacobi-Bellman equation for this reduced model, obtaining IS parameters that we can apply back to the full-dimensional SRN. Analysis and numerical experiments demonstrate that both proposed IS strategies substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.