3. (1) Bob tries to model share value of a spicy peanut butter brand according to a compound Poisson process (R., 120).

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3 1 Bob Tries To Model Share Value Of A Spicy Peanut Butter Brand According To A Compound Poisson Process R 120 1 (46.08 KiB) Viewed 52 times

3. (1) Bob tries to model share value of a spicy peanut butter brand according to a compound Poisson process (R., 120). 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.120) a compound Poisson process that jumps only by +1. Show that E[cºm, ] = ***?-conbu)(2 - 1) sinho), VO E R. for some 2 > 0 and p € (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 a > 0. a) Show with care that in general for any time t > 0, as h→0, P(R+7= R + 1) = Aph + och), P(R+ = R, - 1) = 4(1-P)h + och) and P(R+ = R.) = 1 - Ah + och). b) Denoting p.(t) = P(x, = n), Vn e Z, show that p (t) = -1p.(1) + påpn-1(t) + (1 - p)Apn+i(t). Vt 20,1 € Z. c) Assume that the equation of b) has a unique solution with po(0) = 1 and p.(0) = 0 for N0. Deduce from b) that for all ne Z and 120, P.(I) = where - (+5) (21 XpQ2 –P)e". (1 1, (x) =Σ Σ 5)*", kter,n 20. 21+ 1 7!(1 +n!)