Importance sampling for McKean-Vlasov SDEs

Abstract

This paper deals with Monte-Carlo (MC) methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE) including those with super-linearly growing drifts. Underpinned by an interacting particle system approximation, we propose two importance sampling (IS) algorithms to reduce the variance of an associated MC estimator. The “complete measure change” algorithm sees the IS measure change applied to the target expectation to be calculated and to the MV-SDE coefficients simultaneously. The “decoupling algorithm” consists of estimating the law of the MV-SDE's solution via standard simulation under the initial measure, then fixing the law component of the MV-SDE via that simulation, and finally simulating the new equation under the IS measure. Methodologically, large deviations and Pontryagin principle are employed to determine and solve the variance minimisation problem that yields the required measure change. The optimisation problem associated to the complete measure change is more complex than that for the decoupling algorithm, nonetheless, symmetry arguments allow for non-trivial complexity reduction. As an example, both algorithms are tested using the Kuramoto model from statistical physics. For the functionals tested, we see a reduction of up to 3 orders of magnitude on the variance of both IS schemes in comparison to the standard Monte Carlo approximation. In terms of computational cost, the complete measure change is akin to standard Monte Carlo whilst the decoupled approach increases the cost by a factor of around 2 if one uses the same number of particles for both steps. The statistical error of the method dominates the propagation of chaos error by 1 order of magnitude.