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A 5 points let is a constant hint prove that there

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Unformatted text preview: Problem 4 (19 pts). Let Denote denote a standard Brownian motion. a. (5 points) Let is a constant ∑ ∑ Hint: Prove that there that doesn’t depend on n or a . so that: | |. For the rest of the problem let denote the rational numbers in and for each let the order statistics of be denoted by: [ Example: If then .] b. (5 points) (i). Prove: ∑ bab as ∞. ( ) (ii). How can you improve the result in (i) if there exist ℱ , a decreasing sequence of σ-algebras, so that {∑ ( ) ℱ } is a backwards martingale? c...
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