A Heston local-stochastic volatility model for optimal investment–reinsurance strategy with a defaultable bond in an ambiguous environment

This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment. In other words, the insurer is ambiguous about the insurance claim that is exponentially distributed with an uncertain rate parameter. The insurer can purchase proportional reinsurance and invest its wealth in three assets: a risk-free asset, a risky asset, the price process of which satisfies the Heston local-stochastic volatility model, and a defaultable corporate bond. For the optimal investment–reinsurance objective with a smooth ambiguity utility proposed by Klibanoff, P., Marinacci, M., and Mukerji, S. [A smooth model of decision making under ambiguity, Econometrica, 2005, 73(6): 1849-1892], the equilibrium strategy is introduced and the extended Hamilton–Jacobi–Bellman equation is established through a stochastic control approach. However, the analytical solution of the strategy under the Heston local-stochastic volatility model cannot be obtained because of the complicated nonlinearity of the partial differential equation. In this study, we employ a perturbation method to derive an asymptotic solution for the post- and pre-default cases. In addition, we present a sensitivity analysis to explain the impact of model parameters on the equilibrium investment–reinsurance strategy.