Optimal linear filter for a class of nonlinear stochastic differential systems with discrete measurements

Continuous-discrete models refer to systems described by continuous ordinary or stochastic differential equations, with measurements acquired at discrete sampling instants. Here we investigate the state estimation problem in the stochastic framework, for a class of nonlinear systems characterized by a linear drift and a generic nonlinear diffusion term. Motivation stems from a large variety of applications, ranging from systems biology to finance. By using a Carleman linearization approach we show how the original system can be embedded into an infinite dimensional bilinear system, for which it is possible to write the equations of the optimal linear filter, in case of measurements provided by linear state transformations. A finite dimensional approximation of the optimal linear filter is finally derived. Results are applied to a case of interest in financial applications.