Projected Particle Methods for Solving McKean--Vlasov Stochastic Differential Equations

We propose a novel projection-based particle method for solving 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 situations 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, particularly 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