Stochastic Maximum Principle for a Kind of Risk-sensitive Optimal Control Problem and Application to Portfolio Choice

In this paper, we mainly study a kind of risk-sensitive optimal control problem motivated by a kind of portfolio choice problem in certain financial market. Using the classical convex variational technique, we obtain the maximum principle for this kind of problem. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equation and the variational inequality heavily depend on the risk-sensitive parameter γ. This is one of the main difference from the risk-neutral case. We use this result to solve a kind of optimal portfolio choice problem. The optimal portfolio strategy obtained by the Bellman dynamic programming principle is a special case of our result when the investor only invests the home bond and the stock. Computational results and figures explicitly illustrate the relationships between the maximum expected utility and the parameters of the model.