(i) Bob tries to model share value of a spicy peanut-butter brand according to a compound...

 

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(i) 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 [7n 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[e] = e!\{1=coh)-(2-1)sinh()), 10 e R, for some > 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 > 0. a) Show with care that in general for any time > 0, as h0, = P(Rer R+1)= Aph+o(h), P(Rher R,-1)=(1 - p)h+o(h) = and P(RR) = 1 - Ah+o(h). b) Denoting p,(t) = P(X, n), Vn e Z, show that = P() -p,(t) + papa-1(t) + (1 - p)p+1(t). Vt 0, ne Z. [7n [6 n [5 n (i) 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 [7n 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[e] = e!\{1=coh)-(2-1)sinh()), 10 e R, for some > 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 > 0. a) Show with care that in general for any time > 0, as h0, = P(Rer R+1)= Aph+o(h), P(Rher R,-1)=(1 - p)h+o(h) = and P(RR) = 1 - Ah+o(h). b) Denoting p,(t) = P(X, n), Vn e Z, show that = P() -p,(t) + papa-1(t) + (1 - p)p+1(t). Vt 0, ne Z. [7n [6 n [5 n