Title: Consistency of MLE in partially observed diffusion models.

 

Abstract: Partially observed Markov processes appear in a wide variety of mathematical models, including the latent price models in Finance. When the (static) parameters of such a model are known, the unobserved coordinates of a diffusion process can be approximated based on the observed ones via stochastic filtering. But what shall one do if the parameters of a model also need to be estimated? According to the classical parametric statistics, the maximum likelihood estimator (MLE) is expected to have the best asymptotic properties, as the size of an observed sample (i.e., the length of an observed path) increases. These properties are easy to prove in a basic setting with i.i.d. observations, but establishing them in a more complex model, such as a partially observed Markov model, presents significant challenges. In my talk, I will focus on proving the consistency of MLE — i.e. its convergence to a true parameter value — in partially observed diffusion models with periodic coefficients. This result is new, although analogous results are available in a discrete-time setting. I will discuss the financial problem that served as a motivation for this study and will describe the novel approach to one of the main steps in the proof of consistency, which allowed us to extend it to a diffusion setting.