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

Ning Wang, Nan Zhang, Zhuo Jin, Linyi Qian

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)168-184
Number of pages17
JournalInsurance: Mathematics and Economics
Early online date21 Nov 2020
Publication statusPublished - 1 Jan 2021


  • Dynamic Value-at-Risk (VaR)
  • Nash equilibrium
  • Non-zero-sum stochastic differential game
  • Quadratic risk process
  • Relative performance


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