We introduce a new method for online parameter estimation in stochastic interacting particle systems, based on continuous observation of a small number of particles from the system. Our method recursively updates the model parameters using a stochastic approximation of the gradient of the asymptotic log likelihood, which is computed using the continuous stream of observations. Under suitable assumptions, we rigorously establish convergence of our method to the stationary points of the asymptotic log-likelihood of the interacting particle system. We consider asymptotics both in the limit as the time horizon $t\rightarrow\infty$, for a fixed and finite number of particles, and in the joint limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. Under additional assumptions on the asymptotic log-likelihood, we also establish an $\mathrm{L}^2$ convergence rate and a central limit theorem. Finally, we present several numerical examples of practical interest, including a model for systemic risk, a model of interacting FitzHugh--Nagumo neurons, and a Cucker--Smale flocking model. Our numerical results corroborate our theoretical results, and also suggest that our estimator is effective even in cases where the assumptions required for our theoretical analysis do not hold.

翻译：我们提出了一种基于连续观测系统中少量粒子的随机交互粒子系统在线参数估计新方法。该方法利用观测数据流计算渐近对数似然梯度的随机逼近，递归更新模型参数。在适当假设下，我们严格证明了该方法收敛于交互粒子系统渐近对数似然的驻点。我们同时考虑了固定有限粒子数下时间范围$t\rightarrow\infty$的渐近性，以及粒子数$N\rightarrow\infty$与时间范围$t\rightarrow\infty$的联合极限。在渐近对数似然的附加假设下，我们还建立了$\mathrm{L}^2$收敛速率与中心极限定理。最后，我们展示了多个具有实际意义的数值算例，包括系统性风险模型、交互FitzHugh--Nagumo神经元模型以及Cucker--Smale集群模型。数值结果不仅验证了理论结论，还表明即使在理论分析所需假设不满足的情况下，我们的估计器仍具有良好性能。