ma503a-09-hm3

ma503a-09-hm3 - 5) Assume that a process { X n : n = 1 , 2...

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MA503a (FALL 2009) HOMEWORK #3 Due Wednesday, October 7, 2008 1) Let { B t : t 0 } be a Brownian motion, and {F B t } t 0 be the filtration generated by B . Define M t = B 2 t - t , t 0. Show that M is an {F B t } -martingale. 2) Let X = { X n } n =0 be a martingale with respect to the filtration {F n } n =0 , and τ is a {F n } -stopping time such that τ N , almost surely, for some constant N > 0. For any B ∈ F τ , define τ B = τ 1 B + N 1 B c . Show that τ B is an {F n } -stopping time as well. (Hint: Recall that F τ = { B ∈ F : B ∩ { τ j } ∈ F j , j } , and check { τ B j } ∈ F j , for all j .) 3) Recall that we have proved the “Optional Sampling Theorem” in the following form: If X is an {F n } - martingale, then for all bounded stopping time τ , it holds that E { X τ } = E { X 0 } . Using this fact and Problem 2) to argue that if σ , τ are two bounded stopping times such that σ τ almost surely, then E { X τ |F σ } = X σ , a.s. 4) Using Jensen’s Inequality to argue that i) | E { X }| ≤ E | X | ; ii) | E { X }| 2 E {| X | 2 } , provided that all the expectations involved exist. (A challenge: if you claim that a function is “convex”, you must prove it!)
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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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This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.

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