A stability result for the HARA class with stochastic interest rates

We study an investment problem where the interest rates follow the Cox–Ingersoll–Ross dynamics. The optimal investment strategy is obtained in explicit form under the hypotheses that financial markets are complete and that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solution for the HARA utility is stable when the parameters vary in a suitable way: more precisely, we find that the optimal investment strategy corresponding to the HARA function converges almost surely to the one corresponding to the exponential and logarithmic utility functions.