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. 
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References

SHOWING 1-10 OF 39 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 solveExpand
Approximation of the Zakai¨ equation by the splitting up method
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 stochasticExpand
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 isExpand
Nonlinear filtering : Interacting particle resolution
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 inExpand
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 anExpand
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 approximateExpand
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 beingExpand
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 forExpand
Unique characterization of conditional distributions in nonlinear filtering
  • T. Kurtz, D. 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 theExpand
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 density of a diffusion process observed in white-noise. The case where the observationExpand
...
1
2
3
4
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