Exponential Family Inference for Diffusion Models

We consider ergodic diffusion processes for which the class of invariant measures is an exponential family, and study inference based on the class of invariant probability measures when the diffusion has been observed at discrete time points. When the drift depends linearly on the parameters, the invariant measures form an exponential family. It is investigated how the usual exponential family inference, which can be done by means of standard statistical computer packages, works when the observations are from a diffusion process. In particular, the limit distributions of estimators and test statistics are derived. As an example, we consider classes of diffusions with generalized inverse Gaussian marginals. A particular instance is the well-known Cox-Ingersoll-Ross model from mathematical finance. Key Words: Asymptotic Normality; Consistency; Cox-IngersollRoss model; Discrete time observation; Inference for Diffusion Processes; Estimating Functions; Exponential families; Generalized in..