Online parameter estimation for the McKean-Vlasov stochastic differential equation

We analyse the problem of online parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We propose an online estimator for the parameters of the McKean-Vlasov SDE, or the interacting particle system, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as t & RARR; oo, and also in the joint limit as t & RARR; oo and N & RARR; oo. In these two cases, we obtain almost sure or L1 convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, L2 convergence to the unique maximiser of the asymptotic log-likelihood of the McKean-Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model.& COPY; 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).