A Carleman discretization approach to filter nonlinear stochastic systems with sampled measurements

The state estimation problem, here investigated, regards a class of nonlinear stochastic systems, characterized by having the state model described through stochastic differential equations meanwhile the measurements are sampled in discrete times. This kind of model (continuous-discrete system) is widely used in different frameworks (i.e. tracking, finance and systems biology). The proposed methodology is based on a proper discretization of the stochastic nonlinear system, achieved by means of a Carleman linearization approach. The result is a bilinear discrete-time system (i.e. linear drift and multiplicative noise), to which the Kalman Filter equations (or the Extended Kalman Filter equations in case of nonlinear measurements) can be applied. Because the approximation scheme is parameterized by a couple of indexes, related to the degree of approximation with respect to the deterministic and the stochastic terms, in the numerical simulations, different approximation orders have been used in comparison with standard methodologies. The obtained results encourage the use of the proposed approach.