On a role of predictor in the filtering stability

On the anticipative nonlinear filtering problem and its stability

In this paper, we consider an anticipative nonlinear filtering problem, in which the observation noise is correlated with the past of the signal. This new signal-observation model has its applications in both finance models with insider trading and in engineering. We derive a new equation for the filter in this context, analyzing both the nonlinear and the linear cases. We also handle the case of a finite filter with Volterra type observation. The performance of our algorithm is presented through numerical experiments.

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observati…

Stability of nonlinear filters in nonmixing case

The nonlinear filtering equation is said to be stable if it ``forgets'' the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure.

On exponential stability of Wonham filter

We give elementary proof of a stability result concerning an exponential asymptotic ($t\to\infty$) for filtering estimates generated by wrongly initialized Wonham filter. This proof is based on new exponential bound having independent interest.

Duality for nonlinear filtering

This thesis is concerned with the stochastic filtering problem for a hidden Markov model (HMM) with the white noise observation model. For this filtering problem, we make three types of original contributions: (1) dual controllability characterization of stochastic observability, (2) dual minimum variance optimal control formulation of the stochastic filtering problem, and (3) filter stability analysis using the dual optimal control formulation. For the first contribution of this thesis, a backward stochastic differential equation (BSDE) is proposed as the dual control system. The observabilit…

Uniform observability of hidden Markov models and filter stability for unstable signals

A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition…

关于集成（线性高斯）Kalman-Bucy滤波的数学理论

On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

该综述系统梳理连续时间线性高斯场景下集成Kalman-Bucy滤波的数学理论，分析多类变体的演化方程与收敛、稳定特性，给出适配不稳定潜信号的非渐近结果，并证明自举粒子滤波器无法跟踪基础不稳定潜信号，还讨论了非线性推广与实际滤波行为的关联。