A Class of Non-zero-sum Investment and Reinsurance Games subject to Systematic Risks

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.