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 36 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…
Approximation of the Zakai¨ equation by the splitting up method
- Mathematics, Computer Science
- 1989
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
- 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…
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…
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…
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…
Unique characterization of conditional distributions in nonlinear filtering
- MathematicsThe 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
- 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.