Optimal Consumption by a Bond Investor: The Case of Random Interest Rate Adapted to a Point Process

Abstract

The problem of optimizing the expected total discounted utility of the rate of consumption by an agent with initial wealth x who, at each instant, can choose a rate of consumption and who invests all his unconsumed wealth in a bond (bank) is studied. The bond is assumed to have a randomly varying interest rate with known probabilistic behavior. A general martingale principle is formulated, according to which the optimal consumption rate is expressed in terms of a positive martingale, if one can be found, satisfying an almost sure integral condition. This martingale will be characterized in the case where the stochastic history of the interest rate is adapted to (i.e., expressible in terms of) an underlying counting process. The problem studied can be viewed as a special case of optimal consumption problems in “incomplete markets,” a terminology introduced to the financial literature by Harrison and Pliska [Stochastic Process. Appl., 11 (1981), pp. 215–260].