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["Doob's Principle"] Let (X n , F n ) n=0,1,2,. be a martingale in discrete time with martingale difference sequence Y n = X n X n1 , n = 1, 2, ., Y...

["Doob's Principle"] Let (Xn, Fn)n=0,1,2,... be a martingale in discrete time with martingale difference sequence Yn = Xn − Xn−1, n = 1, 2, . . ., Y0 = X0. Suppose at time n − 1 we place a bet (possibly random in size) of size bn−1 on the forthcoming value Yn, n = 1, 2, . . .. The return from this bet will be bn−1Yn and the total accumulated return at time n is Gn = b0Y1 + b1Y2 + · · · bn−1Yn. Show that (Gn, Fn)n=1,2,... is a martingale, and interpret!

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