A kind of optimal investment problem under inflation and uncertain time horizon

Abstract

In this paper, we study a kind of optimal investment problem under inflation and uncertain time horizon. It can be generally formulated into a stochastic optimal control problem. In particular for the constant relative risk aversion utility, we employ the method of completion of squares to give an explicit form of optimal portfolio and maximum utility by the solution of a stochastic Riccati equation, whose wellposedness is obtained and also of significance in its own right. The most distinguishing result of our work is that the randomness of exit time actually affects not only the optimal portfolio but also the maximum utility in the case of stochastic market parameters. Moreover, we present several numerical examples to show the application of theoretical results and further discuss the influence of inflation and random time horizon from the economic viewpoint.