Parameter and State Estimation of Stochastic Differential Systems

Abstract

The framework of the paper is the use of stochastic differential equations (SDEs) in the modeling of the real world and the subsequent statistical inference problems involving this type of models. We assume that continuous-time observations are available in a specific time interval, although some references on the analogous problems in the case of discrete time observations are included. The following problems are formulated: (i) the parameter estimation problem under full observation of the system/process; (ii) the state estimation of the partially observed system; (iii) the parameter estimation under partial observation of the system, meaning that the state, which is affected by the unknown parameter(s), is not directly observed but it is observed through some observation function with added noise. This last problem is obviously the most difficult of the three and, in some sense, benefits from the solutions or properties found for the other two. Solutions to the formulated problems are discussed, as well as the most important properties. The focus is on the maximum likelihood estimators and on asymptotic properties as the observed time interval goes to infinity or as the observation noise vanishes. The EM algorithm is analyzed, in particular. Finally, the special case of the hypoelliptic SDEs is detailed, and the challenges it brings are discussed. The paper ends with some comments on present and new axes of research around the topic. (c) The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.