A BSDE approach to a class of dependent risk model of mean–variance insurers with stochastic volatility and no-short selling

Abstract

This paper studies the optimal reinsurance and investment strategy for an insurer with two dependent classes of insurance business, where the claim number processes are correlated through a common shock. It is assumed that the insurer also faces the decision making of investing in a financial market with one risk-free asset and one risky asset following the Heston stochastic volatility (SV) model. The insurer is not allowed to short sell the risky asset. Under the mean–variance criterion, we consider the insurer’s problem of maximizing the expected terminal wealth and, at the same time, minimizing the variance of the terminal wealth. Using the results of stochastic linear–quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), we derive closed-form expressions for the optimal strategies and the efficient frontiers in terms of solutions to the BSDEs. Our approach shows how BSDEs can be used to solve mean–variance problems in insurance applications. Finally, economic behavior of the efficient frontiers is analyzed by using some numerical examples.