Parametric estimation for discretely observed systems of stochastic differential equations with common noise

We consider parametric inference in the drift and diffusion terms of N identically distributed diffusion paths with a common noise added to each path. This framework allows to model intrinsic and extrinsic sources of noise. The sample paths are discretely observed over a fixed time interval [0, T ]. The asymptotic framework is that N tends to infinity while the sampling interval T /n tends to 0. The diffusion coefficients are unknown and constant and the drift depends on an unkown parameter. We build a contrast function and prove that the associated maximum contrast estimators are asymptotically normal for the two diffusion parameters, mixed normal for the drift parameter under the conditions that √ n/N and N/n both tend to 0. It results in a non standard parametric inference problem with respectively three ordered rates of convergence for parameters, √ N n, √ n, √ N for the intrinsic diffusion term, common noise diffusion term and drift term respectively. Examples fitting the framework are provided.