Random processes

3. (i) Bob tries to model share value of a spicy peanut-butter brand according to a compound Poisson process (R,, t 2 0). We assume here that the share value of the peanut-butter brand can only change by + 5 cents. After inspection of its historical values, Bob establishes that in expectation the share value changes by +0.3 cents per day and that the standard deviation to this change is 1 cent. Can Bob find a compound Poisson process modelling the situation? (ii) Denote by (R,, t 2 0) a compound Poisson process that jumps only by +1. Show that [[eOR,] = e-ta(1-cosh(0)-(2p-1) sinh(0), VO E R, for some 1 > 0 and p E [0, 1]. (iii) Assume here that the jumps are given by 1 with probability p and -1 with probability 1-p and that the rate of the process is 1 > 0. a) Show with care that in general for any time T > 0, as h - 0, P(RA+T = Ry + 1) = 1ph + o(h), P(RA+T = RT - 1) = 1(1 - p)h + o(h) and P(RA+T = RT) = 1 - Ah + o(h). b) Denoting Pr(t) = P(X, = n), Vn E Z, show that Pr(t) = -Apn(t) + papn-1(t) + (1 - p)ipn+1(t), Vt 2 0, neZ. c) Assume that the equation of b) has a unique solution with po(0) = 1 and Pr (0) = 0 for n # 0. Deduce from b) that for all n E Z and t 2 0, Pn(1 ) = (PIm ( 21 VP(1 - P)) e-, where In(x) = 2 7(1 + n!) (2) ,VIER, n 20. 120