6. (16 marks) Let {X,, n > 0} be a random walk which describes the position of an object at time n that starts at 0 and at each step goes either up h units with probability p or down h units with probability 1 - p. a) 2 Show that if X,, is a standard random walk, i.e. if p = =, then X,, is a martingale. b) 3 Show that if p / ;, then X,, is not a martingale. c) B Let Y, = X - X, 1. Calculate g(A) = E(exYm ). d) 4 Show that Mr. = exxn nloss(A) is a martingale. e) 4 Show that if h = 1, then (1 p) ") , n 2 1, is a martingale