Importance Sampling for McKean-Vlasov Stochastic Differential Equation

Abstract: We are interested in Monte Carlo (MC) methods for estimating probabilities of rare events associated with solutions to the McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs arise in the mean-field limit of stochastic interacting particle systems, which have many applications in pedestrian dynamics, collective animal behaviour and financial mathematics. Importance sampling (IS) is used to reduce high relative variance in MC estimators of rare event probabilities. Optimal change of measure is methodically derived from variance minimisation, yielding a high-dimensional partial differential control equation which is cumbersome to solve. This problem is circumvented by using a decoupling approach, resulting in a lower dimensional control PDE. The decoupling approach necessitates the use of a double Loop Monte Carlo (DLMC) estimator. We further combine IS with a novel multilevel DLMC estimator which not only reduces complexity from O(TOL^-4) to O(TOL^-3) but also drastically reduces associated constant, enabling computationally feasible estimation of rare event probabilities.