Consider p E (0, 1) and a simple walk (Y,),20 on Z with transition matrix P ifm = n + 1 Pnm = 1-p ifm =n- 1 0 otherwise. Let us set Ho = inf{n 2 0 : Y, = 0} be the hitting time of the value a by the chain. a) Show that almost surely, lim In = 2p - 1. 1-00 n b) Let us consider f(y) = Py(Ho , f(1) # {0, 1}. 3/ Show that f (y) = pf(y + 1) + (1 -p)f(y - 1), Vy 2 1