Theory and simulation of interacting particle systems and Mckean-Vlasov processes: the super measure class, ergodicity, and weak error

Abstract

This thesis is divided neatly into four collections of results.

In the first and second parts, we present an implicit Split-Step explicit Euler-type Method (SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in space and then super-linear growth in measure and non-constant Lipschitz diffusion coefficient. In the case that super-linear only happens in space, we prove a classical 1/2 root mean square error (rMSE) convergence rate, for which other schemes have competitive results, but in the case that super-linear happens in space and measure ( in convolution form), the SSM has a near-optimal classical (path-space) rMSE rate of 1/2 − ϵ for ϵ > 0 and an optimal rate 1/2 in the non-path-space MSE, for which the result is new and no other scheme has been proven to work yet. The results are published in [47] and [48].

In the third part, we study a class of MV-SDEs with drifts and diffusions having super-linear growth in measure and space – the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift’s super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient requires novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC). Further, we prove the SSM work in this class of MV-SDEs and attain a rate of 1/2 in the non-path-space MSE, and under further assumptions on the model parameters, we show an exponential ergodicity property for the numerical scheme. The result is published in arXiv [49] and submitted to EJP.

In the fourth part, We study a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the ergodic distribution of a onedimensional McKean–Vlasov Stochastic Differential Equation (MV-SDE) under an assumption of strong convexity (finite-time results are also established). Based on a careful analysis of the variational processes and the backward Kolmogorov equation for the particle system associated to the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (weak convergence rate is 3/2) than the standard Euler method (of weak order 1). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension (N) of the IPS and this includes establishing uniform in time-decay estimates for moments of the IPS, the Kolmogorov equation and their derivatives. We establish several interesting results on the higher-order variation processes of the IPS which are of independent interest. The result will be published. The result submitted to EJP.

We provide different numerical tests of the theory and results for each part.