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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).
 Spring '10
 DURRETT
 The Land

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