Explicit solutions of some utility maximization problems in incomplete markets

In this note we prove Hِlder-type inequalities for products of certain functionals of correlated Brownian motions. These estimates are applied to the study of optimal portfolio choice in incomplete markets when the investorʹs utility is of the form U(X,Y)=g(X)h(Y), where X is the investorʹs wealth and Y is a random factor not perfectly correlated with the market. Explicit solutions are found when g is the exponential, power, or logarithmic utility function.