Show that the adjustment coefficient is log32 if un, Mathematics

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Show that the adjustment coefficient is log32 if un

You are repeatedly playing a game in which at each stage you win 1 with probability 15/19 or lose 2 with probability 4/19.

(a) Show that the adjustment coefficient is log(3/2).

(b) If U_n denotes the amount you have at time n, show that U_n is not a martingale.

(d) Suppose you start with 3 units at time 0. You decide to stop play when you have a total of 20 units, or after 100 plays if that occurs earlier. If US is the amount that you will have after stopping play, what is E((2/3)^U_S )?

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