Projected particle methods for solving McKean-Vlasov stochastic differential equations

TL;DR

This paper introduces a projection-based particle method for efficiently solving McKean-Vlasov stochastic differential equations, reducing computational complexity while providing strong convergence guarantees.

Contribution

It presents a novel projection approach for density estimation in McKean-Vlasov equations, improving efficiency over kernel methods and including convergence analysis.

Findings

Significant reduction in numerical complexity.

Established strong convergence rates.

Validated performance through numerical examples.

Abstract

We propose a novel projection-based particle method for solving the McKean-Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method leads in many situation to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis in the case of linearly growing coefficients turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples.