In this paper, 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.
翻译:在本文中, 我们考虑的是 DE- Vlasov SDE 参数估算的问题, 以及相交的微弱粒子系统的相关系统。 我们首先确定互动粒子系统离线最大可能性估计值的一致性和无症状常态性。 我们首先确定互动粒子系统离线最大可能性估计值的一致性和无症状常态性常态性, 以粒子数量 $N\\rightarrow\ infty$为限。 我们然后提出麦肯那- Vlasov SDE 参数的在线估计值, 该参数根据持续时间的超常性超常性梯度梯度下游算法, 在互动粒子系统失常性对正对正近性对准性。 我们随后证明, 这个估测的离差以$\ mathbr=l=l=l=lg=lgyml=lationalmodeal modeal_ sliyal_lational_lational- laxal- modeal- lax laxal- lax modeal- modeal- modeal- modeal- modeal2, la- modeal- lacial- lacial- modeal- modeal- moudal- moudal- moudal- moudal- moudal2) modal- ta modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal- modal-modal- modal- modal- modal- modal-modal- modal- modal- modal- modal- modal- modal- modal- modal-modal- modal- modal- modal- modal- modal-modal- mod