Double-Loop Importance Sampling for McKean--Vlasov Stochastic Differential Equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with solutions to the d-dimensional McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in [dos Reis et al., 2023], generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of O(TOL−4r) with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.