TY - JOUR

T1 - Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints

AU - Wang, Ning

AU - Zhang, Nan

AU - Jin, Zhuo

AU - Qian, Linyi

N1 - Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2021/1

Y1 - 2021/1

N2 - This paper investigates a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. We allow both insurers to purchase a proportional reinsurance contract and invest in risky and risk-free assets. When applying the generalized mean–variance premium principle in determining reinsurance premium, the surplus process becomes quadratic in the retained proportion of the claims. The optimization criterion of each insurer is to maximize the expected utility of the insurer's terminal performance relative to that of his competitor. In addition, we incorporate dynamic Value-at-Risk (VaR) constraints in the optimization problems of both insurers to satisfy the capital requirements from regulators. The results show that this game problem can be converted to solving a system of nonlinear equations by means of dynamic programming principle and Karush–Kuhn–Tucker (KKT) conditions. Specifically, when both insurers are constant absolute risk aversion (CARA) institutions and the reinsurance premium principle reduces to the expected value principle, we derive the simplified expressions for the Nash equilibrium strategies. Finally, we use some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies under three different scenarios.

AB - This paper investigates a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. We allow both insurers to purchase a proportional reinsurance contract and invest in risky and risk-free assets. When applying the generalized mean–variance premium principle in determining reinsurance premium, the surplus process becomes quadratic in the retained proportion of the claims. The optimization criterion of each insurer is to maximize the expected utility of the insurer's terminal performance relative to that of his competitor. In addition, we incorporate dynamic Value-at-Risk (VaR) constraints in the optimization problems of both insurers to satisfy the capital requirements from regulators. The results show that this game problem can be converted to solving a system of nonlinear equations by means of dynamic programming principle and Karush–Kuhn–Tucker (KKT) conditions. Specifically, when both insurers are constant absolute risk aversion (CARA) institutions and the reinsurance premium principle reduces to the expected value principle, we derive the simplified expressions for the Nash equilibrium strategies. Finally, we use some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies under three different scenarios.

KW - Dynamic Value-at-Risk (VaR)

KW - Nash equilibrium

KW - Non-zero-sum stochastic differential game

KW - Quadratic risk process

KW - Relative performance

UR - http://www.scopus.com/inward/record.url?scp=85097433435&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2020.11.004

DO - 10.1016/j.insmatheco.2020.11.004

M3 - Article

SN - 0167-6687

VL - 96

SP - 168

EP - 184

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

ER -