Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s SV model

This paper considers an optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s stochastic volatility (SV) model. Suppose that the insurer is allowed to purchase excess-of-loss reinsurance and invests her surplus in a financial market consisting of one risk-free asset and one risky asset whose price process is described by Heston’s SV model. Under the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic differential delay equation. The insurer’s aim is to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. By adopting the dynamic programming approach, the optimal strategies and the optimal value functions are derived explicitly under two cases: the investment-reinsurance case and the investment-only case. Finally, some numerical examples and sensitivity analysis are provided for our results.