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

Abstract

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.

Using systematic sampling selection for Monte Carlo solutions of Feynman-Kac equations

“…The Bernoulli sampling selection method was introduced in [10]. See also [6] and [8] for additional properties of this sampling selection method.…”

“…The Bernoulli sampling selection method was introduced in [10]. See also [6] and [8] for additional properties of this sampling selection method. It is worth noting that M i n takes the same values as in the systematic sampling selection method, provided that N n−1 = N. For n ≥ 1, and given ξ n−1 and…”

Simultaneous multi-information fusion and parameter estimation for robust 3-D indoor positioning systems

“…Besides, particle filters are also used by some researchers to solve the similar problem. One problem using particle filters is that when the parameter is part of state, the augmented state space model is not ergodic, and the uniform convergence result does not hold anymore [15].…”

On the convergence of the wavelet-Galerkin method for nonlinear filtering

“…There are a lot of methods of constructing approximations of the Zakai equation: splitting up method (Elliott and Glowinski, 1989), decomposition into Wiener integrals (Crisan et al, 1998), discrete time approximations (Bennaton, 1985), some generalizations in Hilbert spaces (Germani and Picconi, 1984). There are also some modifications of the Galerkin method and various bases are used, e.g., the Gaussian series basis, in the paper (Ahmed and Radaideh, 1997).…”