Long-Term Investment with Stochastic Interest and Inflation Rates Incompleteness and Compensating Variation

We examine the long term investment problem, under stochastic interest and inflation rates and incompleteness. Four basic financial assets are available on the financial market: a money market account (the cash), a real consumption good, a financial stock index and a bond with constant maturity. This one corresponds to a nominal bond. In this incomplete framework, we provide the general solution of the expected utility maximization. This intertemporal optimization problem is solved by using the convex duality method, introduced by Cvitanic and Karatzas (1992). We determine also the optimal portfolio weights by using the method of dynamic programming based on the Hamilton-Jacobi-Bellmann approach. We compute the monetary loss from not having access to an indexed-inflation bond, in order to be hedged against the inflation risk, in particular for the logarithmic case.