停时σ代数上两概率测度等价性的一点注记

A NOTE OF EQUIVALENCE OF TWO PROBABILITY MEASURES ON STOPPING TIME σ ALGEBRA

设P,Q为可测空间(Ω,F)上的两概率测度,(F_■)_(■∈N)为F的单调上升的子α代数序列。J.Yeh在他的论著中用J-散度给出了在F_∞=σ(∩■F_n)上两概率测度等价的一个充分条件。本短文将这个结果推广到对任意(F_■)_(■∈N)停时v,在v前σ代数F_■上仍成立。同时对连续参数集合R_+=[0,∞),在一定的条件下亦有类似的结论。

Let(Ω, F) be a measurable space and P、Q probability measures on F. (F_n)_(n∈N) is an increasing family of sub-σ-algebras of F. J. Yeh gives a sufficient condition for equivalence of P and Q on F_∞=σ(■ F_n) in terms of J-divergence in the book written by him. We extend this result to F_v where v is a stopping time with respect to (F_n)_(n∈N). Meanwhiln under certain conditions similar conclusions hold for continuous parameter set R=[0, ∞].