Exercise 13.2.6 Let { Xn } be a martingale and letCn denote the stake on the nth game. Cn may depend on X1, X2, . . . , Xn−1 and is bounded. C1 is a constant. Interpret Cn(Xn − Xn−1) as the gains on the nth game. The total gains up to game n are Yn ≡
n i=1 Ci (Xi − Xi−1) with Y0 = 0. Prove that {Yn } is a martingale with respect to { In }, where In ≡ {X1, X2, . . . , Xn }.