A branching particle system approximation for nonlinear stochastic filtering

The optimal filter 7r = {π,t ∈ [0, T]} of a stochastic signal is approximated by a sequence {Try} of measure-valued processes defined by branching particle systems in a random environment （given by the observation process）. The location and weight of each particle are governed by stochastic differential equations driven by the observation process, which is common for all particles, as well as by an individual Brownian motion, which applies to this specific particle only. The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n-2α, where n is the number of initial particles and ~ is a fixed parameter to be optimized. As n → ∞, we prove the convergence of π to πt uniformly for t ∈ [0, T]. Compared with the available results in the literature, the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.