# Nonlinear filtering and measure-valued processes

@article{Crisan1997NonlinearFA, title={Nonlinear filtering and measure-valued processes}, author={Dan Crisan and Terry Lyons}, journal={Probability Theory and Related Fields}, year={1997}, volume={109}, pages={217-244} }

Summary. We construct a sequence of branching particle systems with time and space dependent branching mechanisms whose expectation converges to the solution of the Zakai equation. This gives an alternative numerical method to solve the Filtering Problem.

## 82 Citations

### Central limit theorem for nonlinear filtering and interacting particle systems

- Mathematics
- 1999

Several random particle systems approaches were recently suggested to solve nonlinear ltering problems numerically. The present analysis is concerned with genetic-type interacting particle systems.…

### Central Limit Theorem for Non Linear Filtering and Interacting Particle Systems

- Mathematics
- 1999

Several random particle systems approaches were recently suggested to solve numerically non linear ltering problems. The present analysis is concerned with genetic-type interacting particle systems.…

### Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering

- Mathematics
- 2000

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The…

### Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems

- Mathematics
- 1998

In the paper we study interacting particle approximations of discrete time and measure-valued dynamical systems. These systems have arisen in such diverse scientic disciplines as physics and signal…

### Particle Approximations for a Class of Stochastic Partial Differential Equations

- Mathematics
- 2006

The paper presents a particle approximation for a class of nonlinear stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The new results permit the…

### Covariance identities and mixing of random transformations on the Wiener space

- Mathematics
- 2010

In this paper we derive criteria for the mixing of random trans- formations of the Wiener space. The proof is based on covariance identities for the Hitsuda-Skorokhod integral.

### Convergence of a Branching Particle Method to the Solution of the Zakai Equation

- MathematicsSIAM J. Appl. Math.
- 1998

A sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation is constructed, which can be used to solve numerically the filtering problem.

### Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering

- Mathematics
- 2000

This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology,…

### Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics

- Mathematics
- 2000

In this paper we formulate a numerical approximation method for the nonlinear filtering of vortex dynamics subject to noise using particle filter method. We prove the convergence of this scheme…

### Large deviations for interacting particle systems: Applications to non-linear filtering

- Mathematics, Computer Science
- 1998

## References

SHOWING 1-10 OF 13 REFERENCES

### A criterion of convergence of measure‐valued processes: application to measure branching processes

- Mathematics
- 1986

In this paper martingale properties of a Measure Branching process are investigated. Uniqueness and continuity of this process are proven by a martingale approach. For the existence, we approximate…

### Approximation of nonlinear filtering problems and order of convergence

- Mathematics
- 1984

In this paper, we consider a filtering problem where the observation is a function of a diffusion corrupted by an independent white noise. We estimate the error caused by a discretization of the time…

### Convergence of Probability Measures

- MathematicsThe Mathematical Gazette
- 1970

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

### Stochastic Control of Partially Observable Systems

- Mathematics, Computer Science
- 1992

This paper presents a meta-analysis of stochastic control methods for linear dynamic systems with quadratic payoff and some explicit solutions of the Zakai equation.

### Topics in Nonlinear Filtering Theory.

- Mathematics
- 1980

Abstract : This thesis studies two topics in the theory of nonlinear filtering, the use of multiple stochastic integrals to analyze filters, and the use of Lie algebraic and operator-theoretic…

### New Results in Linear Filtering and Prediction Theory

- Mathematics
- 1961

The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results and properties of the variance equation are of great interest in the theory of adaptive systems.

### Stopping Times and Tightness. II

- Mathematics
- 1978

To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of…

### Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images

- PhysicsIEEE Transactions on Pattern Analysis and Machine Intelligence
- 1984

The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.

### A new approach to linear filtering and prediction problems" transaction of the asme~journal of basic

- Engineering
- 1960

A unitary, lightweight outer garment constructed of a thin polyethylene film includes front and rear panels which are joined together forming a medial body member, paired arms which extend outwardly…