Double-loop importance sampling for McKean–Vlasov stochastic differential equation

Statistics and Computing, vol. 34

Abstract

Abstract This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution 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 $$P \times d$$ 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 $$\mathcal {O}\left( \textrm{TOL}_{\textrm{r}}^{-4}\right) $$ O TOL r - 4 with a significantly reduced constant to achieve a prescribed relative error tolerance $$\textrm{TOL}_{\textrm{r}}$$ TOL 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 $$\textrm{TOL}_{\textrm{r}}$$ TOL r compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Authors 4

1. Nadhir Ben Rached

   University of Leeds

   Affiliation as printed

   Department of Statistics, School of Mathematics, University of Leeds, Leeds, UK

2. Abdul-Lateef Haji-Ali

   Heriot-Watt University

   Affiliation as printed

   Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK

3. Shyam Mohan Subbiah Pillai corresponding Aachen Chair of Mathematics for Uncertainty Quantification Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, RWTH Aachen University, Aachen, Germany

4. Raul F. Tempone Aachen

   King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Aachen, Germany

   Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia