Optimal investment, consumption, and work effort strategies with stochastic salary under the HLSV model

We consider a Heston local-stochastic volatility (HLSV) model to study the optimal investment, consumption, and work effort strategies for an executive with equity incentives and stochastic salary. The executive’s work behavior can affect the stock price of the company, and after retirement, he can choose whether to accept re-employment. Our goal is to maximize the expected discounted utility of consumption and terminal wealth and reduce the loss utility generated by work. In addition, we establish the Hamilton–Jacobi–Bellman (HJB) equation through the dynamic programming principle and solve complex nonlinear partial differential equations using a perturbation method to obtain asymptotic solutions for the optimal strategies and value function under the HLSV model. We also consider a general case in which the price process of risky asset follows the stochastic volatility (SV) model, which can be compared with the HLSV model. Finally, we provide a numerical example to illustrate the impact of some important parameters on the optimal strategies and value function.