Estimating Continuous-Time Models with Discretely Sampled Data

Abstract

INTRODUCTION Since Merton's seminal work in the 1970s, the continuous-time paradigm has proved to be an immensely useful tool in finance and more generally economics. Continuous time models are widely used to study issues that include the decision to optimally consume, save, and invest; portfolio choice under a variety of constraints; contingent claim pricing; capital accumulation; resource extraction; game theory and more recently contract theory. The objective of this lecture is to review some of the developments in the econometric literature devoted to the estimation and testing of these models. The unifying theme of the class of the problems I will discuss is that the data generating process is assumed to be a continuous-time process describing the evolution of state variable(s), but the process is sampled, or observed, at discrete time intervals. The issues that arise, and the problems that are of interest, at the interface between the continuous-time model and the discrete-time data are quite different from those that we typically encounter in standard time series analysis. As a result, there has been a large amount of research activity in this area. I will start with the simplest possible model, under many assumptions that restrict its generality, and describe how different inference strategies can be developed to work under progressively richer settings, where I relax either some aspect of the model's specification and/or the manner in which the process is sampled.

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