Time-consistent catastrophe risk management under the path-dependent effects

This paper investigates optimal investment and insurance strategies under a mean-variance criterion with path-dependent effects. We use a rough volatility model with a power kernel and a Hawkes process with a modified Omori kernel (a power kernel) to capture the market's path dependence. By extending the functional Ito calculus to the mixed fractional Brownian-Hawkes process, we derive the corresponding path-dependent extended Hamilton-Jacobi-Bellman equation and solve its explicit solution. For numerical analysis, we first calibrate the Hawkes process with the Wenchuan earthquake data, the most devastating earthquake in China. We find that the power kernel outperforms the exponential one in fitting the earthquake intensity. Our numerical results reveal that the path-dependent effect strongly depends on the horizon length. For the investment strategy, the individual is more risk-seeking when considering the rougher volatility in the short horizon, but more risk-averse at the beginning of the long horizon. For the insurance strategy, a faster decay in intensity increases individuals' demand for catastrophe insurance in the short horizon, but decreases initially in the long horizon. However, we find that the horizon effect may disappear when the shift parameter in the modified Omori kernel approaches zero or exceeds one. Our findings indicate that ignoring path-dependent effects would lead to significant underinsurance and highlight its importance in catastrophe risk management.