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

2018 ◽  
Vol 50 (2) ◽  
pp. 396-413
Author(s):  
Huijie Qiao
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Filtering ◽  
General Assumption ◽  
Observation System ◽  
Pathwise Uniqueness ◽  
Zakai Equation ◽  
Stochastic Dynamical Systems ◽  
Gaussian Signal ◽  
Non Gaussian ◽  
Filtering Problem

Abstract In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. 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.

Nonlinear filtering of stochastic dynamical systems with Lévy noises

2015 ◽  
Vol 47 (03) ◽  
pp. 902-918 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Filtering ◽  
Martingale Problem ◽  
Strong Solutions ◽  
Observation System ◽  
Regularity Conditions ◽  
Signal System ◽  
Zakai Equation ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian

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.

Nonlinear filtering of stochastic dynamical systems with Lévy noises

2015 ◽  
Vol 47 (3) ◽  
pp. 902-918 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Filtering ◽  
Martingale Problem ◽  
Strong Solutions ◽  
Observation System ◽  
Regularity Conditions ◽  
Signal System ◽  
Zakai Equation ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian

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.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

scholarly journals On the convergence of the wavelet-Galerkin method for nonlinear filtering

2010 ◽  
Vol 20 (1) ◽  
pp. 93-108 ◽  
Author(s):  
Łukasz Nowak ◽  
Monika Pasławska-Południak ◽  
Krystyna Twardowska
Keyword(s):  
Galerkin Method ◽  
Compact Support ◽  
Nonlinear Filtering ◽  
Euler Scheme ◽  
Zakai Equation ◽  
Implicit Euler Scheme ◽  
Galerkin Scheme ◽  
Wavelet Galerkin Method ◽  
Biorthogonal Basis ◽  
Filtering Problem

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise

2019 ◽  
Vol 17 (03) ◽  
pp. 477-511 ◽  
Author(s):  
Shenglan Yuan ◽  
Jianyu Hu ◽  
Xianming Liu ◽  
Jinqiao Duan
Keyword(s):  
Biological Sciences ◽  
Dynamical Systems ◽  
Scale Parameter ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Time Dynamics ◽  
Slow Manifolds ◽  
Long Time ◽  
Non Gaussian ◽  
Levy Noise

This work is concerned with the dynamics of a class of slow–fast stochastic dynamical systems driven by non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, and then we eliminate the fast variables to reduce the dimensions of these stochastic dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps to investigate long time dynamics. The approximations of slow manifolds with error estimate in distribution are also established. Furthermore, we corroborate these results by three examples from biological sciences.

scholarly journals Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Amirhossein Taghvaei ◽  
Jana de Wiljes ◽  
Prashant G. Mehta ◽  
Sebastian Reich
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Nonlinear Problems ◽  
Improved Stability ◽  
Consistent Solution ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Filtering Problem

This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman–Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman–Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman–Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

Asymptotic methods for stochastic dynamical systems with small non-Gaussian Lévy noise

2014 ◽  
Vol 15 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Asymptotic Methods ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Nonlocal Interactions ◽  
Escape Probabilities ◽  
Non Gaussian

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric α-stable Lévy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e. differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.

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