Answer Happy

1. Recall the asymmetric random walk problem. We are given the process (Xn) with Xo = a > 0, defined via Xn+1 = Xn + (2Bn – 1), for B1, B2, ... ~iid Ber(p). We define the stopping time (b> a > 0) T = min{n : {Xn = 0} U{Xn =b}}. Show that P(T < 0) = 1. =

S 2. Show that the time-reversed process {W 3,5 € [0, t]} defined by W, = Wi-s – Wų is a Wiener process on (0,t]. 3. If {W+} is a Wiener process, show that {Xt} with X+ = Wat/va, t> 0 for any a > 0 is also a Wiener process.