Information theory for maximum likelihood estimation of diffusion models

We develop an information theoretic framework for maximum likelihood estimation of diffusion models. Two information criteria that measure the divergence of a diffusion process from the true diffusion are defined. The maximum likelihood estimator (MLE) converges asymptotically to the limit that minimizes the criteria when sampling interval decreases as sampling span increases. When both drift and diffusion specifications are correct, the criteria become zero and the inverse curvatures of the criteria equal the asymptotic variance of the MLE. For misspecified models, the minimizer of the criteria defines pseudo-true parameters. Pseudo-true drift parameters depend on approximate transition densities if used.