26. Suppose that A dollars are invested in a bank that pays interest at a rate of X per year, where X is a random variable.

(a) Show that if a year is divided into k equal periods, and the bank pays interest at the end of each of these k periods, then after n such periods, with probability 1, the investment will grow to

(b) For an infinitesimal ε > 0, suppose that the interest is compounded at the end of each period of length ε. If ε → 0, then the interest is said to be compounded continuously. Suppose that at time t, the investment has grown to A(t). By demonstrating that A(t + ε) = A(t) + A(t) · εX, show that, with probability 1, A′(t) = XA(t).

(c) Using part (b), prove that, If the bank compounds interest continuously, then, on average, the money will grow by a factor ofMX(t), the moment-generating function of the interest rate.

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A(1+)