映射族的推移算子及其在随机过程中的应用

Shift Operators of Map-Family and Its Applicationto Stochastic Processes

该文研究很一般化的映射族的推移算子 .设参数集 T=[0 ,+∞ )或 T={0 ,1 ,2 ,… },以及任意非空的集合 Ω和 E,Ω和 E均不配置可测 σ代数 ,更不配置测度或概率 .在引进 δ映射和 δ代数的基础上 ,研究了 Ω到 E的映射族 X={X( t) ,t∈ T∩ [0 ,σ) }( σ∈ T∪ {+∞ })存在推移算子的充分必要条件 .讨论了推移算子的诸多性质 。

The authors researched the shift operators of map family in vague generalization. Given a parameter set T=[0,+∞) or T={0,1,2,...}, and two arbitrary non empty sets Ω and E,which are not disposed any measurable σ algebra and any measure.The set of all maps φ: T∩[0,β)→E ,where β∈T∪{+∞}, is denoted by Ω (E). The map X: Ω→Ω(E) is a map family X={X(t),t∈T∩[0,σ)} in fact. It is followed that the necessary and sufficient condition of existence for shift operators of a map family X={X(t),t∈T∩[0,σ)}. Some properties of shift operators and some applications to stochastic processes for shift operators are discussed.