In this problem we will study the convergence of the growing rate of an investment. Let say that we can invest some money in two ways l A bond that always returns a 6% increase each year a A stock that increases your investment by 50% with a probability {3.8 or decreases your investment by 1UU% with a probability 0.2 We assume that each year is independent of the previous one and that those two possibilities never change. We also assume that you can put any proportion of your money in either investment 1 or in investment 2. The goal is of course to maximize your prot. At the beginning of the process you start with In dollars and invest the proportion p (D 1: p E. 1) in the stock and 1 p in the bond. We can define a random variable I, that is equal to 1 if the stock goes up and is equal to 0 if the stock goes down for the year n.(e) Show that ilog X\" 3\" 1og(1.' -|- (1.44;?) + (1 &)log[1.[1 p}] In it n (f)I Use the convergence in probi':i.'|:.~i_1it:,r to compute ,1. in function of j:- (g) Find the optimal value of p to optimize the A found in (f)