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. 
View via Publisher
stat.purdue.edu
94 Citations
Highly Influential Citations
6
Background Citations
22
Methods Citations
30
Results Citations
4

Figures from this paper

94 Citations

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

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…

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

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.

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

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…

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

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

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

Particle representations for a class of nonlinear SPDEs

An efficient Monte Carlo scheme for Zakai equations

It is shown that under suitable regularity assumptions on the coefficients of the Zakai equation the corresponding random PDE admits a solution random which can be written as a classical solution of a second order linear parabolic PDE.

Numerical Solutions for a Class of SPDEs with Application to Filtering

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.

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

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…

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

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…
...

References

SHOWING 1-10 OF 36 REFERENCES

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

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…

Approximation of the Zakai¨ equation by the splitting up method

An operator splitting method is applied to the time integration of Zakai equation to decompose the numerical integration into a stochastic step and a deterministic one, both of them much simpler to handle than the original problem.

Approximation of some stochastic differential equations by the splitting up method

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 : Interacting particle resolution

Nonlinear filtering and measure-valued processes

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…

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

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…

Discretization and simulation of stochastic differential equations

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…

Nonlinear Filtering Revisited: A Spectral Approach

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…

Unique characterization of conditional distributions in nonlinear filtering

  • T. KurtzD. Ocone
  • 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…

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

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.