Importance sampling methods for McKean-Vlasov type stochastic differential equations

Subbiah Pillai, Shyam Mohan; Tempone, Raul (Thesis advisor); Haji-Ali, Abdul-Lateef (Thesis advisor); Ben Rached, Nadhir (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Master Thesis

Masterarbeit, RWTH Aachen University, 2022

Abstract

In this thesis, we are interested in Monte Carlo methods for estimating probabilities of rare events associated with solutions to the McKean-Vlasov stochastic differential equation (MV-SDE), whose drift and diffusion coefficients depend on the law of the solution itself. The MV-SDE is approximated using the stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations. Importance sampling is used to reduce high variance in Monte Carlo estimators of rare event probabilities. Optimal change of measure is methodically derived from variance minimization, yielding a (P x d)-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 estimator. In this context, we formulate an adaptive double loop Monte Carlo method for estimating rare event probabilities. Significant variance reduction is observed and the computational runtime for estimating rare event probabilities up to a given relative tolerance, TOL, is reduced by multiple orders, when compared to standard Monte Carlo estimators. We also formulate a novel multilevel double loop Monte Carlo (MLDLMC) method combined with importance sampling, to estimate rare events in the MV-SDE context. This reduces the order of optimal work complexity from O(TOL^{-4}) to O(TOL^{-3}). Our numerical experiments are carried out on the Kuramoto model from statistical physics, which models a system of coupled oscillators.