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Unformatted text preview: > n) (q/p)a Px (⌧ > n)
Letting n ! 1 we have established (5.11). We can save ourselves some work by abstracting the last argument. Theorem 5.14. Suppose Mn is a martingale and T a stopping time with P (T <
1) = 1 and MT ^n  K for some constant K . Then EMT = EM0 .
Proof. Theorem 5.13 implies EM0 = EMT ^n = E (MT ; T n) + E (Mn ; T > n).
The second term KP (T > n) and
E (MT ; T n) E (MT ) KP (T > n) Since P (T > n) ! 0 as n ! 1 the desired result follows.
Example 5.11. Duration of fair games. Let Sn = S0 + X1 + · · · + Xn
where X1 , X2 , . . . are independent with P (Xi = 1) = P (Xi = 1) = 1/2. Let
⌧ = min{n : Sn 62 (a, b)} where a < 0 < b. Our goal here is to prove a close
relative of (1.26):
E0 ⌧ = ab
2
Example 5.4 implies that Sn n is a martingale. Let ⌧ = min{n : Sn 62 (a, b)}.
From the previous example we have that ⌧ is a stopping time with P (⌧ < 1) =
2
1. Again if we argue casually 0 = E0 (S⌧ ⌧ ) so using (5.10)
2
E0 (⌧ ) = E0 (S⌧ ) = a2 P0 (S⌧ = a) + b2 P0 (S⌧ = b)
b
a
ab
= a2
+ b2
= ab...
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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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