Expected power utility maximization with delay for insurers under the 4/2 stochastic volatility model

Abstract

In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer with delay under the $ 4/2 $ stochastic volatility model. Indeed, the objective of the insurer is to maximize the expected power utility of the terminal wealth and the average wealth on finite time horizon. The other objective is to maximize the growth rate of expected power utility per unit time on infinite time horizon. The wealth of the insurer is described by an approximation of the classical Cramér–Lundberg process. Then, these problems can be formulated as stochastic control problems with delay. A pair of forward-backward stochastic differential equations that are derived via the stochastic maximum principle has an explicit solution obtained by solving a Riccati differential equation. So, the optimal strategy of the finite time horizon problem can be constructed explicitly. And, by investigating asymptotics of the Riccati equation, the infinite time horizon problem can be solved explicitly. Finally, we present some numerical results to illustrate our model, optimal strategies and sensitivities of some parameters.