Optimal asset allocation with heterogeneous discounting and stochastic income under CEV model

This article focuses on an optimal asset allocation problem with heterogeneous discounting and stochastic income under the constant elasticity of variance (CEV) model. Heterogeneous discounting is an important non-constant discounting model, which can describe the fact that a decision maker discounts in different ways the utility derived from consumption and that of the bequest or final function. It is consistent with the fact that the concern of a decision maker about the bequest left to her descendants when she is young is not the same as that when she is old. In our model, a decision maker with stochastic income can enjoy the consumption, purchase life insurance, and invest her wealth in a risk-free asset and a risky asset whose price process satisfies the CEV model. Meanwhile, the volatility of the stochastic income arises from the risky asset. Since the problem is time-inconsistent, the Bellman's principle of optimality does not hold. To obtain the time-consistent solution, an equilibrium strategy is calculated. By applying the game theoretic framework and solving an extended Hamilton-Jacobi-Bellman system, we derive the time-consistent consumption, investment, and life insurance strategies for both exponential and logarithmic utility functions. Finally, we provide numerical simulations to illustrate our results.