Compound Poisson particle approximation for McKean–Vlasov SDEs

We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha $-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean–Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean–Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes our approximation exhibits universality. In the case of ordinary differential equations (ODEs) we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application we construct a compound Poisson approximation for two-dimensional Navier–Stokes equations on the torus and demonstrate the optimal convergence rate.