It is easy to see that is a stopping time lemma 13

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Unformatted text preview: 5 4 10+1 3 10+4 2 5+6 1 1+4+1 0 3+1 so our net winnings are 0 with probability 52/64 = 0.8125. The negative values, though less frequent, are larger 1 3 2 1 4 1 5 2 6 1 13 1 14 1 30 1 63 1 Adding everything up we see that our expected winnings = 145/64 145/64 = 0. To formulate and prove (5.7) we will introduce a family of betting strategies that generalize the doubling strategy. The amount of money we bet on the nth game, Hn , clearly, cannot depend on the outcome of that game, nor is it sensible to allow it to depend on the outcomes of games that will be played later. We say that Hn is an admissible gambling strategy or predictable process if for each n the value of Hn can be determined from Xn 1 , Xn 2 , . . . , X0 , M0 . To motivate the next definition, think of Hm as the amount of stock we hold between time m 1 and m. Then our wealth at time n is Wn = W0 + n X Hm (Mm Mm 1) (5.8) m=1 since the change in our wealth from time m 1 to m is the amount we hold times the change in the price of the stock: Hm (Mm Mm 1 ). To formulate the doubling strategy in this...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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