Optimal investment, consumption, and life insurance strategies under a mutual-exciting contagious market

We study an optimisation problem of a household under a contagious financial market. The market consists of a risk-free asset, multiple risky assets and a life insurance product. The clustering effect of the market is modelled by mutual-excitation Hawkes processes where the jump intensity of one risky asset depends on both the jump path of itself and the jump paths of other risky assets in the market. The labor income is generated by a diffusion process which can cover the Ornstein-Uhlenbeck (OU) process and the Cox-Ingersoll-Ross (CIR) model. The goal of the household is to maximise the expected utilities from both the instantaneous consumption and the terminal wealth if he survives up to a fixed retirement date. Otherwise, a lump-sum heritage will be paid. The mortality rate is modelled by a linear combination of exponential distributions. We obtain the optimal strategies through the dynamic programming principle and develop an iterative scheme to solve the value function numerically. We also provide the proof of convergence of the iterative method. Finally, we present a numerical example to demonstrate the impact of key parameters on the optimal strategies.