Q. 1. (20 pts) Let (12,F,P) be a probability space with a discrete-time filtration (Fn)nezt. Let (Sn)nez, be a submartin

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Q. 1. (20 pts) Let (12,F,P) be a probability space with a discrete-time filtration (Fn)nezt. Let (Sn)nez, be a submartin

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Q 1 20 Pts Let 12 F P Be A Probability Space With A Discrete Time Filtration Fn Nezt Let Sn Nez Be A Submartin 1
Q 1 20 Pts Let 12 F P Be A Probability Space With A Discrete Time Filtration Fn Nezt Let Sn Nez Be A Submartin 1 (75.65 KiB) Viewed 33 times
Q. 1. (20 pts) Let (12,F,P) be a probability space with a discrete-time filtration (Fn)nezt. Let (Sn)nez, be a submartingale with respect to (Fn)nezt. (a) (10 pts) Show that there exists a martingale (Mn)nezt with respect to (Fn)nezt and an increasing predictable process (An)nen such that Sn = Mn + An = for each n E Z+. (Take Ao O as a convention.) Hint: Consider the “mismatch” in the submartingale property and define (An)neN accordingly. (b) (10 pts) Show that the decomposition in (a) is unique almost surely. It is called the Doob decomposition of (Sn)nezt.
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