## Abstract

Recently, there have been numerous insightful applications of zero-sum

stochastic differential games in insurance, as discussed in Liu et al. [Liu,

J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer

with regime-switching and risk constraint. Scandinavian Actuarial Journal

2014(7), 583–601]. While there could be some practical situations under

which nonzero-sum game approach is more appropriate, the development

of such approach within actuarial contexts remains rare in the existing

literature. In this article, we study a class of nonzero-sum reinsuranceinvestment

stochastic differential games between two competitive insurers

subject to systematic risks described by a general compound Poisson risk

model. Each insurer can purchase the excess-of-loss reinsurance tomitigate

both systematic and idiosyncratic jump risks of the inter-arrival claims;

and can invest in one risk-free asset and one risky asset whose price

dynamics follows the famous Heston stochastic volatility model [Heston,

S. L. (1993). A closed-form solution for options with stochastic volatilitywith

applications to bond and currency options. Review of Financial Studies 6,

327–343]. The main objective of each insurer is to maximize the expected

utility of his terminal surplus relative to that of his competitor. Dynamic

programming principle for this class of nonzero-sum game problems leads

to a non-canonical fixed-point problem of coupled non-linear integraltyped

equations. Despite the complex structure, we establish the unique

existence of the Nash equilibrium reinsurance-investment strategies and

the corresponding value functions of the insurers in a representative

example of the constant absolute risk aversion insurers under a mild,

time-independent condition. Furthermore, Nash equilibrium strategies and

value functions admit closed forms. Numerical studies are also provided

to illustrate the impact of the systematic risks on the Nash equilibrium

strategies. Finally, we connect our results to that under the diffusionapproximated

model by proving explicitly that the Nash equilibrium

under the diffusion-approximated model is an -Nash equilibrium under

the general Poisson risk model, thereby establishing that the analogous

Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam,

S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential

investment and reinsurance games. Automatica 50(8), 2025–2037] serves

as an interesting complementary case of the present framework.

stochastic differential games in insurance, as discussed in Liu et al. [Liu,

J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer

with regime-switching and risk constraint. Scandinavian Actuarial Journal

2014(7), 583–601]. While there could be some practical situations under

which nonzero-sum game approach is more appropriate, the development

of such approach within actuarial contexts remains rare in the existing

literature. In this article, we study a class of nonzero-sum reinsuranceinvestment

stochastic differential games between two competitive insurers

subject to systematic risks described by a general compound Poisson risk

model. Each insurer can purchase the excess-of-loss reinsurance tomitigate

both systematic and idiosyncratic jump risks of the inter-arrival claims;

and can invest in one risk-free asset and one risky asset whose price

dynamics follows the famous Heston stochastic volatility model [Heston,

S. L. (1993). A closed-form solution for options with stochastic volatilitywith

applications to bond and currency options. Review of Financial Studies 6,

327–343]. The main objective of each insurer is to maximize the expected

utility of his terminal surplus relative to that of his competitor. Dynamic

programming principle for this class of nonzero-sum game problems leads

to a non-canonical fixed-point problem of coupled non-linear integraltyped

equations. Despite the complex structure, we establish the unique

existence of the Nash equilibrium reinsurance-investment strategies and

the corresponding value functions of the insurers in a representative

example of the constant absolute risk aversion insurers under a mild,

time-independent condition. Furthermore, Nash equilibrium strategies and

value functions admit closed forms. Numerical studies are also provided

to illustrate the impact of the systematic risks on the Nash equilibrium

strategies. Finally, we connect our results to that under the diffusionapproximated

model by proving explicitly that the Nash equilibrium

under the diffusion-approximated model is an -Nash equilibrium under

the general Poisson risk model, thereby establishing that the analogous

Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam,

S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential

investment and reinsurance games. Automatica 50(8), 2025–2037] serves

as an interesting complementary case of the present framework.

Original language | English |
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Pages (from-to) | 670-707 |

Number of pages | 38 |

Journal | Scandinavian Actuarial Journal |

Volume | 2017 |

Issue number | 8 |

DOIs | |

Publication status | Published - 15 Sept 2017 |

Externally published | Yes |