Equilibrium excess-of-loss reinsurance–investment strategy for a mean–variance insurer under stochastic volatility model

ABSTRACT

This article considers an optimal excess-of-loss reinsurance–investment problem for a mean–variance insurer, and aims to develop an equilibrium reinsurance–investment strategy. The surplus process is assumed to follow the classical Cramér–Lundberg model, and the insurer is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Under the mean–variance criterion, equilibrium reinsurance–investment strategy and the corresponding equilibrium value function are derived by applying a game theoretic framework. Finally, numerical examples are presented to illustrate our results.

KEYWORDS:

• Equilibrium strategy
• excess-of-loss reinsurance
• mean–variance criterion
• square-root model
• stochastic volatility model.

MATHEMATICS SUBJECT CLASSIFICATION:

• 91B28
• 91B30
• 91B70

Funding

This research was supported by the National Natural Science Foundation of China under Grant Nos. (11301376, 71573110).

Notes

1 As shown in Chen et al. (2014) and Sun et al. (2016), for a dynamic optimization problem, if the strategy is optimal for the decision-maker at some time t₁, and for any later time t₂ > t₁, she will follow the strategy because it is still optimal at time t₂, i.e., for all t > t₂, then it is called a time-consistent strategy. Under the game theory framework, the derived subgame perfect Nash equilibrium strategy is a time-consistent strategy, see, for example, Björk and Murgoci (2010) and Björk et al. (2014).