一个无穷质点系统的滤波(鞅方法)

FILTERING FOR AN INFINITE PARTICLES SYSTEM:A MARTINGALE APPROACH

(一)记I^0=(1,2,…)表示所考虑的质点系统可能到达的“位置”集,φ表示I^0中有限子集全体.假设:(I)外界质点按强度λ_i>0的泊松过程(At(j))_t≥0独立于输入“位置”j∈I^0且λ≡∑λ_j<∞;(Ⅱ)质点一旦到达j之后,各自独立地按同一马氏链X^0运动,且与(A_t(j))独立;设诸质点寿命是相互独立与一正r.v.L同分布。

Consider an infinite particles system that admited to the particles randomly input a coun-table set I, and then they move in I according to the transition law of a known Markov chain.we show that the inno J?I; Let finite subset vations methed and Martingale theory is used toderive the filtering equations satisfied by the conditional probability distribution of the stateon J when the past of an observed set of the output stream was given, and these filtering equa-tions are applied to obtain a necessary and sufficient condition for that observation to be in-dependent of the value of the state of the system on J at anytime. In particular, It is discussedto the system in equilibrium behaviour.