Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.