η = j1 = = ө Exercise 4.11. Again let S be the random walk Sn = }=1 X; with {X;}; i.i.d., taking ==; , values +1 with p

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η = j1 = = ө Exercise 4.11. Again let S be the random walk Sn = }=1 X; with {X;}; i.i.d., taking ==; , values +1 with p > 1/2 and So = 0. Let u(0) = Ee0X1, VO E R. Show that for any 0 ER, the process Mn := eosn/u(0)" is a martingale w.r.t. En := 0(X1, X2, .. , Xn), Fo {0, 1}. Moreover, let T = inf{n > 0: Sn = 1}. Note that t < 0 a.s. Show that for ~ 0 > 0, u(0) > 1 and Eu(0)-7 = e-o. [5 marks] п := = > . -T =