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

@article{Crisan1998ConvergenceOA, title={Convergence of a Branching Particle Method to the Solution of the Zakai Equation}, author={Dan Crisan and Jessica G. Gaines and Terry Lyons}, journal={SIAM J. Appl. Math.}, year={1998}, volume={58}, pages={1568-1590} }

We construct a sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation. The algorithm based on this result can be used to solve numerically the filtering problem. The result is an improvement of the one presented in a recent paper [Crisan and T. Lyons, Prob. Theory Related Fields, 109 (1997), pp. 217--244], because it eliminates the extra degree of randomness introduced there.

#### 92 Citations

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

- Mathematics
- 1998

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… Expand

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

- Mathematics
- 2003

In Crisan, Gaines and Lyons [SIAM J. Appl. Probab. 58 (1998) 313--342] we describe a branching particle algorithm that produces a particle approximation to the solution of the Zakai equation and find… Expand

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… Expand

Convergence of a branching and interacting particle system to the solution of a nonlinear stochastic PDE

- 2004

|The solution of a nonlinear parabolic SPDE on the circle, with multiplicative Gaussian noise that is white-noise in time and a bona ̄de function in space, is approximated by a system of branching… Expand

Convergence of a branching and interacting particle system to the solution of a nonlinear stochastic PDE

- Mathematics
- 2004

The solution of a nonlinear parabolic SPDE on the circle, with multiplicative Gaussian noise that is white-noise in time and a bonaflde function in space, is approximated by a system of branching and… 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

Particle representations for a class of nonlinear SPDEs

- Mathematics
- 1999

An innite system of stochastic dierential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V, and V… Expand

Numerical Solutions for a Class of SPDEs with Application to Filtering

- Mathematics
- 2001

A simulation scheme for a class of nonlinear stochastic partial differential equations is proposed and error bounds for the scheme are derived. The scheme is based on the fact that the solutions of… 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

A Branching Particle Approximation to the Filtering Problem with Counting Process Observations ∗

- 2006

Recently, the filtering model with counting process observations has been demonstrated as a sensible framework for modeling the micromovement of asset price (or ultra-high frequency data). In this… Expand

#### References

SHOWING 1-10 OF 39 REFERENCES

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

- Mathematics
- 1998

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… Expand

Approximation of the Zakai¨ equation by the splitting up method

- Mathematics
- 1989

The objective of this article is to apply an operator splitting method to the time integration of Zakai equation. Using this approach one can decompose the numerical integration into a stochastic… Expand

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… Expand

Nonlinear filtering : Interacting particle resolution

- Mathematics
- 1997

Abstract In this Note, we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as in… Expand

Nonlinear filtering and measure-valued processes

- 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… Expand

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… Expand

Discretization and simulation of stochastic differential equations

- Mathematics
- 1985

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being… Expand

Nonlinear Filtering Revisited: A Spectral Approach

- Mathematics
- 1997

The objective of this paper is to develop an approach to nonlinear filtering based on the Cameron--Martin version of Wiener chaos expansion. This approach gives rise to a new numerical scheme for… Expand

Unique characterization of conditional distributions in nonlinear filtering

- Mathematics
- The 23rd IEEE Conference on Decision and Control
- 1984

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… Expand

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

- Mathematics
- 1991

A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional density of a diffusion process observed in white-noise. The case where the observation… Expand