Constrained equilibrium investment and risk control strategies under a non-Markovian regime-switching model

This article addresses optimal investment and risk control strategies within a non-Markovian regime-switching model under the mean-variance framework. It assumes that the returns and volatility matrix of risky assets in the financial market, as well as the parameters of the insurance risk model, are non-Markovian processes, adapted to the filtration generated by a Markov chain. The introduction of the Markov chain models the impact of changes in the external macroeconomic environment on the market. Insurers aim to minimize a mean-variance utility function by adjusting policy issuance volumes (the risk control strategy) and investing the surplus in the financial market (the investment strategy). This study considers a risk aversion coefficient that depends on the current wealth level, indicating a state-dependent utility function. Furthermore, it assumes that strategies prohibit short-selling and require the wealth process to satisfy a non-bankruptcy constraint. Based on the ideas of the stochastic maximum principle (SMP), the equilibrium strategies are described through forward-backward stochastic differential equations (FBSDEs) with two variational inequalities representing equilibrium conditions. By decoupling this complex system, the article explicitly presents the open-loop equilibrium strategies in a projected form, based on solutions to a set of coupled regime-switching BSDEs. Finally, we discuss a specific scenario of the model, namely, the Markovian regime-switching model, and elaborate on the theoretical results through numerical examples.