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

Abstract

In the article, Zakai and Kushner-Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system are considered. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner-Stratonovich equation is unique in joint law.

Uniqueness and superposition of the distribution-dependent Zakai equations

“…Nonlinear filtering problems of the systems like (1) are called as nonlinear filtering problems of McKean-Vlasov SDEs with correlated noises. And then we deduce the Kushner-Stratonovich equation ( 4), the Zakai equation (10) and the distributiondependent Zakai equation (12) about the system (1). Next, we view Eq.…”

“…Nonlinear filtering problems are to extract some useful information of unobservable phenomenon from observable ones and then estimate and predict these unobservable phenomenon (c.f. [1,8,9,10,11,12,13,15,16]). And nonlinear filtering theory plays an important role in many areas including stochastic control, financial modeling, speech and image processing, and Bayesian networks.…”

Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme

“…Moreover, nonlinear dynamics is also a factor that needs to be considered. In [13], Zakai and Kushner-Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system are considered. Operator theory is a kind of nonlinear control theory which has the characteristics of nonlinear and uncertain unstable elements [14][15][16][17][18][19][20].…”

Nonlinear filtering of stochastic differential equations driven by correlated Lévy noises

“…Therefore, we make use of martingale problems and the Kallianpur-Striebel formula to obtain the Kushner-Stratonovich and Zakai equations. About uniquenesses for the two equations, there are two methods-a filtered martingale problem ( [2,18]) and an operator equation ( [17]). Specially, in [18], Prof. Duan and the author required that λ in the driving processes of the observation process Y t is independent of x.…”

“…Specially, in [18], Prof. Duan and the author required that λ in the driving processes of the observation process Y t is independent of x. And we assumed that the driving processes of the signal process X t has no jumps term in [17].…”