Nonlinear filtering of stochastic dynamical systems with Lévy noises

Abstract

Nonlinear filtering is investigated in a system where both the signal system and the observation system are under non-Gaussian Lévy fluctuations. Firstly, the Zakai equation is derived, and it is further used to derive the Kushner-Stratonovich equation. Secondly, by a filtered martingale problem, uniqueness for strong solutions of the Kushner-Stratonovich equation and the Zakai equation is proved. Thirdly, under some extra regularity conditions, the Zakai equation for the unnormalized density is also derived in the case of α-stable Lévy noise.

Uniqueness for measure-valued equations of nonlinear filtering for stochastic dynamical systems with Lévy noise

“…The equation satisfied byP t (F ) is called the Zakai equation. Based on Theorem 3.1 of [8], we have the following result.…”

“…Thus, using a similar approach to that in [8], we know that −1 t is an exponential martingale. Define a probability measureP via dP dP = −1 T .…”

“…In this subsection we observe the nonlinear filtering problem for a non-Gaussian signalobservation system, and state the Zakai and Kushner-Stratonovich equations; see [8].…”

“…It is worthwhile to mention that in [8], Qiao and Duan proved pathwise uniqueness for the Zakai equation. There the property was defined for strong solutions.…”

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative $α$-stable noises

“…A much cleaner approach, but just as lengthy, is to recast the Zakai equation in its strong form. It is also worth mentioning that the paper [35] presented the strong form of Zakai equation under additive α-stable noise.…”