LAMN property for drift inference in systems of stochastic differential equations with common noise

We consider parametric inference in the drift term of a system of N identically distributed diffusion paths where a common noise is added to each path. This results in correlated identically distributed diffusions paths. The diffusion matrix of the system is supposed to be known. The purpose of this paper is to analyze the effects of a common noise on the statistical inference. The sample paths are continuously observed on a finite time interval [0,T]. The asymptotic framework is N tends to infinity. Having proved that the diffusion matrix of the system is invertible with explicit inverse, we study the exact likelihood. We prove that the statistical model has a limiting random information matrix (LAMN property). The exact maximum likelihood estimator is proved to be consistent and to converge with rate N to a mixture of Gaussian distributions. We compare the results to those obtained when there is no common noise (N i.i.d. sample paths). Various examples with constant and non constant diffusion terms including the Cox-Ingersoll-Ross and the bilinear processes are provided.