Unique characterization of conditional distributions in nonlinear filtering

@article{Kurtz1984UniqueCO,
  title={Unique characterization of conditional distributions in nonlinear filtering},
  author={Thomas G. Kurtz and Daniel Ocone},
  journal={The 23rd IEEE Conference on Decision and Control},
  year={1984},
  pages={693-698}
}
  • T. KurtzD. Ocone
  • Published 1 December 1984
  • Mathematics
  • The 23rd IEEE Conference on Decision and Control
A 'filtered' martingale problem is defined for the problem of estimating a process X from observations of Y, where (X,Y) is Markov. We give conditions on the generator of (X,Y) that imply that the conditional distribution is the unique solution to this filtered martingale problem. We apply this result to prove uniqueness of solutions of the Kushner-Stratonovich and Zakai equations of non-linear filtering. 
View on IEEE
projecteuclid.org
123 Citations
Highly Influential Citations
24
Background Citations
33
Methods Citations
40
Results Citations
3

123 Citations

Stationary Solutions and Forward Equations for Controlled and Singular Martingale Problems

Stationary distributions of Markov processes can typically be characterized as probability measures that annihilate the generator in the sense that for ; that is, for each such , there exists a…

On the Innovations Conjecture of Nonlinear Filtering with Dependent Data

We establish the innovations conjecture for a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The approach uses only elementary stochastic…

Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering

We consider the Zakai equation for the unnormalized conditional distribution $\sigma$ when the signal process $X$ takes values in a complete separable metric space $E$ and when $h$ is a continuous,…

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

  • H. Qiao
  • Mathematics
    Advances in Applied Probability
  • 2018
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…

Conjecture of Nonlinear Filtering with Dependent Data ∗

We establish the innovations conjecture for a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The approach uses only elementary stochastic…

Nonlinear Filtering for Jump Diffusion Observations

We deal with the filtering problem of a general jump diffusion process, X, when the observation process, Y, is a correlated jump diffusion process having common jump times with X. In this setting, at…

A separation principle for partially observed control of singular stochastic processes

Martingale Problems for Conditional Distributions of Markov Processes

Let $X$ be a Markov process with generator $A$ and let $Y(t)=\gamma (X(t))$. The conditional distribution $\pi_t$ of $X(t)$ given $\sigma (Y(s):s\leq t)$ is characterized as a solution of a filtered…

On Uniqueness of Solutions for the Stochastic Differential Equations of Nonlinear Filtering

We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The main results establish pathwise uniqueness for the unnormalized filter equation and…

The filtered martingale problem

Let X be a Markov process characterized as the solution of a martingale problem with generator A, and let Y be a related observation process. The conditional distribution t of X(t) given observations…
...

References

SHOWING 1-2 OF 2 REFERENCES

Multidimensional Diffusion Processes

Preliminary Material: Extension Theorems, Martingales, and Compactness.- Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure.- Parabolic Partial Differential Equations.- The…

Stochastic differential equations for the non linear filtering problem

The general nonlinear filtering or estimation problem may be described as follows. xty (0<t<T)y called the signal or system process is a stochastic process direct observation is not possible. The…