A state predictor for continuous-time stochastic systems

This work investigates the state prediction problem for nonlinear stochastic differential systems, affected by multiplicative state noise. This problem is relevant in many state-estimation frameworks such as filtering of continuous-discrete systems (i.e.~stochastic differential systems with discrete measurements) and time-delay systems. A very common heuristic to achieve the state prediction exploits the numerical integration of the deterministic nonlinear equation associated to the noise-free system. Unfortunately these methods provide the exact solution only for linear systems. Instead here we provide the exact state prediction for nonlinear system in terms of the series expansion of the expected value of the state conditioned to the value in a previous time instant, obtained according to the Carleman embedding technique. The truncation of the infinite series allows to compute the prediction at future times with an arbitrary approximation. Simulations support the effectiveness of the proposed state-prediction algorithm in comparison to the aforementioned heuristic method.