Math 3589 Introduction to Financial Mathematics Homework Assignment #11 1. Prove that a symmetric random walk is a martingale. 2. Prove that a symmetric random walk is a Markov process. 2 3. Prove that the first passage time, m , m = 0, 1, 2, . . . is a stopping time. 3 4. Consider the asymmetric random walk with probability P(H) = p > 1/2 and q = 1 p < 1/2. Let 1 be the first time the random walk reaches the level 1. If the random walk never reaches the level 1, then 1 = . (a) Define f () = pe + qe . Show that f () > 1 for all > 0. 4 (b) Show that for > 0, the process defined by Sn = eMn f n () is a martingale. 5 (c) Show that for > 0, \u0003 \u0002 E0 I1 < f 1 () = e . 6 (d) Why did part (c) imply P[1 < ] = 1? 7 5. Consider the symmetric random walk, and let 2 be the first time the random walk reaches the level 2. Use the reflection principle to show that for integers k = 1, 2, 3, . . . \u0015 \u0012 \u00132k2 (2k 2)! (2k 2)! 1 P[2 = 2k] = + (k 1)!(k 1)! (k)!(k 2)! 2 \u0015 \u0012 \u00132k \u0014 (2k)! 1 (2k)! + (k)!(k)! (k + 1)!(k 1)! 2 \u0014 (you may notice that if n is odd, P[2 = n] = 0.) 8