Unformatted text preview: 5) Assume that a process { X n : n = 1 , 2 , ···} satisﬁes the following: ∀ ε > 0, ∃ δ > 0, such that E { X n  : A } < ε, whenever P ( A ) < δ. Show that lim K →∞ sup n E { X n  1 { X n  >K } } = 0. Namely { X n } is uniformly integrable . 6) Using Problem 5) to show that if { X n } is uniformly integrable, then (i) sup n E ([ X n ] + ) < ∞ ; (ii) If we assume further that { X n } is a martingale, then lim n X n = X ∞ exists (by upcrossing theorem), and E  X ∞  < ∞ (Hint: use Fatou). 7) Assume that Z is a random variable with E  Z  < ∞ and {G n } is a ﬁltration. Show that M n := E { Z G n } is an U.I. {G n }martingale. 8)* Use all the problems above, prove Problem 3) again. But this time assume only that σ and τ are a.s. ﬁnite (i.e., P { σ < ∞} < ∞ , P { τ < ∞} < ∞ ). 1...
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 Fall '09
 MAJIN
 Brownian Motion, Probability theory, Xn, Martingale, Fτ

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