# A particle approximation of the solution of the Kushner–Stratonovitch equation

@article{Crisan1998APA, title={A particle approximation of the solution of the Kushner–Stratonovitch equation}, author={Dan Crisan and Terry Lyons}, journal={Probability Theory and Related Fields}, year={1998}, volume={115}, pages={549-578} }

Abstract. We construct a sequence of branching particle systems αn convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve numerically the filtering problem. We prove that the rate of convergence of the algorithm is of order n¼. This paper is the third in a sequence, and represents the most efficient algorithm we have identified so far.

## 102 Citations

### 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.

### Interacting particle systems approximations of the Kushner-Stratonovitch equation

- MathematicsAdvances in Applied Probability
- 1999

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

### Exact rates of convergeance for a branching particle approximation to the solution of the Zakai equation

- Mathematics, Computer Science
- 2003

The exact rate of convergence of the mean square error is deduced and several variations of the branching algorithm with better rates of convergence are introduced.

### 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,…

### A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems

- Mathematics
- 2004

In an earlier paper, we studied the approximation of solutions V (t) to a class of SPDEs by the empirical measure V n (t) of a system of n interacting difiusions. In the present paper, we consider a…

### Numerical Solutions for a Class of SPDEs with Application to Filtering

- Mathematics, Computer Science
- 2001

A simulation scheme for a class of nonlinear stochastic partial differential equations is proposed and error bounds for the scheme are derived and the results can be applied to nonlinear filtering problems.

### Numerical solutions for a class of SPDEs over bounded domains

- Mathematics
- 2007

The optimal filter for a bounded signal with reflecting boundary is approximated by the (un-weighted) empirical measure of a finite interacting particle system. The main motivation of this…

### An Ensemble Kushner-Stratonovich-Poisson Filter for Recursive Estimation in Nonlinear Dynamical Systems

- MathematicsIEEE Transactions on Automatic Control
- 2016

A Monte Carlo filter for recursive estimation of diffusive processes that modulate the instantaneous rates of Poisson measurements is proposed, with a key aspect the additive filter-update scheme, which eliminates the problem of particle collapse encountered in many conventional particle filters.

### Approximate McKean–Vlasov representations for a class of SPDEs

- Mathematics
- 2005

The solution of a class of linear stochastic partial differential equations is approximated using Clark's robust representation approach. The ensuing approximations are shown to coincide with the…

### Numerical solution for a class of SPDEs over bounded domains

- Mathematics
- 2014

Parabolic stochastic partial differential Equations (SPDEs) with multiplicative noise play a central rôle in nonlinear filtering. More precisely, the conditional distribution of a partially observed…

## References

SHOWING 1-10 OF 10 REFERENCES

### 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.

### Approximation of some stochastic differential equations by the splitting up method

- Mathematics
- 1992

In this paper we deal with the convergence of some iterative schemes suggested by Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic equation is…

### Nonlinear filtering and measure-valued processes

- Business, Mathematics
- 1997

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…

### Time-discretization of the Zakai equation for diffusion processes observed in correlated noise

- Computer Science, Mathematics
- 1990

A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise, and an error estimate of order √β is proved for the overall numerical scheme.

### Nonlinear Filtering Using Random Particles

- Mathematics
- 1996

This paper is concerned with extending the particle solution of nonlinear discrete-time filtering problems developed in [Ph.D. thesis, Universite Paul Sabatier, Tolouse, France, 1994], [Contrat…

### 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…

### 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 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.