In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval $[0, T]$, we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($\Delta_n$) and the number of particles ($N$) satisfy $\Delta_n \rightarrow 0$ and $N \rightarrow \infty$, and asymptotically normal when additionally the condition $\Delta_n N \rightarrow 0$ holds.
翻译:在本文中,我们考虑的是关于Stochastic McKan-Vlasov方程式的漂移和扩散系数以及相联的交互粒子系统的联合参数估计问题。 分析是在一个总的框架内提供的, 因为这两个系数都取决于过程的解决方案和解决方案本身的规律。 从固定间隔[ $0, T]$对互动粒子系统的离散观测开始, 我们根据假可能性方法提出一个对比函数。 我们显示, 当离散步骤($\ Delta_ n$)和粒子数量($)满足 $\ Delta_ n\rightrow 0$和$\rightrow\infty$时, 相关的估计值是一致的, 当附加条件($\ Delta_n N\rightrower 0$) 时, 则不那么正常。