8. (Power utility) A stock price is governed by ds S where is a standardized Wiener process. Interest is constant at rate. An investor wishes to construct a constantly rebalanced portfolio of these two assets that maximizes the expected value of his power utility U(X) = (1/y)X", y < 1, at all times C. Show that the proportion of wealth invested in the risky asset is w = (-)/(1 - y)]. Use the following steps

(a) Show that 11 X(r) = X(0)+1/2+ ==== where is a normal random variable with mean 0 and variance I

(b) Use E() to show that E[U (X())] =

(c) Find w 7/22/2