Identification of continuous-time linear filters when only discrete-time data is available

We study the inference of continuous-time linear filters from discrete-time observations of the underlying stochastic differential equation. This problem poses several identification issues related to the existence of infinite continuous-time specifications with different covariance structures having the same exact discrete-time representation. We derive mild restrictions on the drift dynamics, allowing to uniquely determine the drift linear projections and other relevant covariance structures of the stochastic differential equation. Our results are employed to identify, based on discrete-time data, the solution to an optimal stochastic control problem under partial information, thereby reconciling the continuous-time formulation of the problem with the statistical inference of the model parameters based on discrete-time information. We illustrate, through an empirical application using high-frequency financial data, how our identification scheme enhances the expected utility of intraday trading strategies.