Multiperiod Securities Markets With Differential Information Martingales And R
Multiperiod Securities Markets With Differential Information Martingales And R
Darrell Duffie
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1. Let B e h^{S). Then B G h^a{S). Proof. Define f ^ : fi ^ [0, oo] by = oo Vw ^ B. It is clear that f§ = M{te[0, l]:E{lB\hf{S)) = l}. Thus Tg is an if" (S)-optional random variable. It then follows from Theorem IV. 53 of Dellacherie and Meyer [1982] that B G h^^{S). I Remark 6. 1: For the definitions of optional random variables and hj-a{S), see Chung and Williams [1983, Section 1. 7]. Lemma 6. 2. If H^{S) C H°'{S), then any H^{S)-optional random variable is H^iSyoptional. Furthermore, let T b...e H^[S)-optional. Then h^{S) C h^{S). Proof. Let T be /r''(S)-optional. Then we have {T
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