# 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.

#### 78 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.… Expand

Central Limit Theorem for Non Linear Filtering and Interacting Particle Systems

- 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.… Expand

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… Expand

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… Expand

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… Expand

Interacting Particle Systems Approximations ofthe

- 1999

In this paper we consider the continuous time ltering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will… Expand

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

- Computer Science, Mathematics
- SIAM 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. Expand

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,… Expand

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

- Mathematics
- 1998

The non-linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be… Expand

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