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

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 $$P \times d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>×</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> -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 $$\mathcal {O}\left( \textrm{TOL}_{\textrm{r}}^{-4}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mtext>TOL</mml:mtext> <mml:mrow> <mml:mtext>r</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msubsup> </mml:mfenced> </mml:mrow> </mml:math> with a significantly reduced constant to achieve a prescribed relative error tolerance $$\textrm{TOL}_{\textrm{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>TOL</mml:mtext> <mml:mtext>r</mml:mtext> </mml:msub> </mml:math> . 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 $$\textrm{TOL}_{\textrm{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>TOL</mml:mtext> <mml:mtext>r</mml:mtext> </mml:msub> </mml:math> compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.