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. 
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References

SHOWING 1-10 OF 10 REFERENCES
Convergence of a Branching Particle Method to the Solution of the Zakai Equation
TLDR
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
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 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
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. The case where the observation noiseExpand
Nonlinear Filtering Using Random Particles
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], [ContratExpand
Stopping Times and Tightness. II
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 ofExpand
Stochastic Control of Partially Observable Systems
Preface 1. Linear filtering theory 2. Optimal stochastic control for linear dynamic systems with quadratic payoff 3. Optimal control of linear stochastic systems with an exponential-of-integralExpand
A criterion of convergence of measure-valued processes: Application to measure branching processes
On etudie les proprietes de martingale d'un processus ramifie de mesure. On demontre l'unicite et la continuite de ce processus par une approche de martingale