Some remarks on pramarts and mils

Wang, Zhen-Peng ; Xue, Xing-Hong

Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 239-251

Résumé

Let F be a Banach space, (ω, F, P) a fixed probability space, D a directed set filtering to the right with the order ≤, and (Ft, D) a stochastic basis of F, i.e. (Ft, D) is an increasing family of sub-σ-algebras of F:Fs ⊂ for any s,t ε D and s≤t. Throughout this paper, (Xt) is an F-valued, (Ft)-adapted sequence, i.e. Xt, is Ft-measurable, t ε D. We also assume that Xt, ∈ L1, i.e. ∫ ∥Xt∥ <∞. We use I(H) to denote the indicator function of an event H. Let ∞ be a such element: t <∞, t ∈ D, = D ∪ ∞, and F∞ = σ. A stopping time is a map τ:Ω→ such that (τ<t) ∈ Ft, t ∈ D. A stopping time τ is called simple (countable) if it takes finitely (countably) many values in D(). Let T and Tc be the sets of simple and countable stopping times respectively and Tf = {τ ∈ Tc: τ<∞ a.s.}. Clearly, (T, <) and (Tf, <) are directed sets filtering to the right. For τ ∈ Tc, letand= {(Xt): there is σ∈ Tf such that ∫(ι<∞) ∥Xι∥ < ∞, σ ≤ τ ∈ Tc},= {(Xt):(Xι, ι ∈ T) converges stochastically (i.e. in probability) in the norm topology},E = {(Xt):(Xι, ι ∈ T) converges essentially in the norm topology}.

Détail

DOI : 10.1017/S0017089500009800

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