Shyam Pillai (King Abdullah University of Science and Technology (KAUST)) – Estimating rare-event probabilities associated with the McKean--Vlasov equation

This talk addresses the efficient Monte Carlo estimation of rare-event probabilities associated with a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs), which arise in the analysis of mean-field systems in statistical physics, mathematical finance, and collective behaviour models. Standard Monte Carlo methods become computationally infeasible in this setting due to the rapid growth of the estimator's relative variance (coefficient of variation) in the rare-event regime. Using stochastic optimal control, an optimal importance sampling measure change is constructed to minimise the variance of the resulting estimator. The resulting double-loop Monte Carlo (DLMC) estimator with importance sampling significantly mitigates this growth in the coefficient of variation. The framework is further extended to the multilevel Monte Carlo setting to reduce computational complexity, leveraging propagation-of-chaos and strong antithetic coupling to ensure that the level differences vanish in the mean-field limit. To address the discontinuity of the probability observable, a numerical smoothing technique is introduced to recover optimal variance convergence rates.  Numerical experiments on linear mean-field, Kuramoto, and Cucker--Smale models demonstrate computational savings of several orders of magnitude compared with standard Monte Carlo.