9 to check the last equality consider two cases i if

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Unformatted text preview: Av ) 0. The result in Theorem 5.9 generalizes immediately to our other two types of processes. Multiplying by 1 we see: Theorem 5.10. If Mm is a submartingale and 0 m < n, then EMm EMn . Since a process is a martingale if and only if it is both a supermartingale and submartingale, we can conclude that: Theorem 5.11. If Mm is a martingale and 0 m < n then EMm = EMn . The most famous result of martingale theory (see Theorem 5.12) is that “you can’t beat an unfavorable game.” (5.7) To lead up to this result, we will analyze a famous gambling system and show why it doesn’t work. Example 5.7. Doubling strategy. Suppose you are playing a game in which you will win or lose $1 on each play. If you win you bet $1 on the next play but if you lose then you bet twice the previous amount. The idea behind the system can be seen by looking at what happens if we lose four times in a row and then win: outcome bet net profit L 1 1 L 2 3 L 4 7 L 8 15 W 16 1 In this example our net profit when we win is $1. Since 1+2+ · ·...
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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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