Multilevel importance sampling for rare events associated with the McKean–Vlasov equation

Statistics and Computing, vol. 35

Abstract

Abstract This work combines multilevel Monte Carlo with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean–Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in Ben Rached et al. (Stat Comput, 2024. https://doi.org/10.1007/s11222-024-10497-3 ) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in Ben Rached et al. (2024), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-4})}$$ O ( TOL r - 4 ) for the single-level DLMC estimator to $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-3})}$$ O ( TOL r - 3 ) while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $$\textrm{TOL}_{\textrm{r}}$$ TOL r .

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 Department of Mathematics Chair of Mathematics for Uncertainty Quantification

   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

   RWTH Aachen University · King Abdullah University of Science and Technology

   Affiliation as printed

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

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