These walks are called left continuous since they

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Unformatted text preview: e values of M0 , X0 , X1 , . . . , Xm 1 . Having introduced the gambling strategy “Bet $1 on each play up to time T ” our next step is to compute the payo↵ we receive when W0 = M0 . Letting T ^ n denote the minimum of T and n, i.e., it is T if T < n and n if T n, we can give the answer as: W n = M0 + n X Hm (Mm Mm m=1 1) = MT ^n (5.9) To check the last equality, consider two cases: (i) if T n then Hm = 1 for all m n, so Wn = M0 + (Mn M0 ) = Mn (ii) if T n then Hm = 0 for m > T , so the sum in (5.9) stops at T . In this case, Wn = M0 + (MT M0 ) = MT Combining (5.9) with Theorem 5.12 and using Theorem 5.9 we have Theorem 5.13. If Mn is a supermartingale with respect to Xn and T is a stopping time then the stopped process MT ^n is a supermartingale with respect to Xn . In particular, EMT ^n M0 As in the discussion after Theorem 5.9, the analogous results are true for submartingales (EMT ^n M0 ) and for martingales (EMT ^n = M0 ). 168 CHAPTER 5. MARTINGALES 5.4 Applications In this section we wi...
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