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

TL;DR

This paper develops an efficient double-loop importance sampling method for estimating rare-event probabilities in McKean-Vlasov SDEs, reducing computational complexity and variance through a decoupling approach and adaptive algorithms.

Contribution

It introduces a novel double-loop Monte Carlo estimator with importance sampling for McKean-Vlasov SDEs, leveraging a decoupling approach to improve computational efficiency and accuracy.

Findings

Achieves optimal complexity of O(TOL_r^{-4}) with reduced constants.

Significantly reduces variance and runtime compared to standard Monte Carlo methods.

Demonstrates effectiveness on the Kuramoto model from statistical physics.

Abstract

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 $P \times d$-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 $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ with a significantly reduced constant to achieve a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. 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 $\mathrm{TOL}_{\mathrm{r}}$ compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.