We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.

翻译：我们研究了基于单个粒子连续观测的随机相互作用粒子系统的递归最大似然估计。在该设定下，即使粒子数$N\rightarrow\infty$且时间范围$t\rightarrow\infty$，有限粒子对数似然的一致估计仍然不可行。因此，我们寻求优化极限均场系统的平稳对数似然。本文通过连续时间中的随机梯度估计形式实现了这一目标，其中随机梯度估计利用单个被观测粒子的连续轨迹，并结合虚拟相互作用粒子系统及虚拟切向相互作用粒子系统进行计算，这些系统与在线参数估计同步更新。对于固定的真实与虚拟粒子数，我们证明所提算法驱动有限粒子替代目标函数的梯度在$t\to\infty$时趋于零。进一步证明，在极限顺序$t\to\infty$后取$N,M\to\infty$时，这些替代梯度一致收敛于极限均场系统平稳对数似然的梯度，从而收敛至其驻点。我们通过多个数值算例验证了该方法，包括含二次约束和相互作用势的模型、相互作用的FitzHugh-Nagumo神经元模型，以及随机Kuramoto模型。