Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation

We consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We first establish consistency and asymptotic normality of the offline maximum likelihood estimator for the interacting particle system in the limit as the number of particles $N\rightarrow\infty$. We then propose an online estimator for the parameters of the McKean-Vlasov SDE, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. We prove that this estimator converges in $\mathbb{L}^1$ to the stationary points of the asymptotic log-likelihood of the McKean-Vlasov SDE in the joint limit as $N\rightarrow\infty$ and $t\rightarrow\infty$, under suitable assumptions which guarantee ergodicity and uniform-in-time propagation of chaos. We then demonstrate, under the additional assumption of global strong concavity, that our estimator converges in $\mathbb{L}^2$ to the unique maximiser of this asymptotic log-likelihood function, and establish an $\mathbb{L}^2$ convergence rate. We also obtain analogous results under the assumption that, rather than observing multiple trajectories of the interacting particle system, we instead observe multiple independent replicates of the McKean-Vlasov SDE itself or, less realistically, a single sample path of the McKean-Vlasov SDE and its law. Our theoretical results are demonstrated via two numerical examples, a linear mean field model and a stochastic opinion dynamics model.