Timeconsistent nonzerosum stochastic differential reinsurance and investment game under default and volatility risks Abstract   This paper investigates a nonzerosum stochastic differential game between two meanxe2x80x93variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a riskfree asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The nonzerosum goal of these insurers is to maximize the meanxe2x80x93variance utility of a weighted value of their terminal and relative wealths. We solve the meanxe2x80x93variance problem in the timeconsistent case and establish the extended Hamilton   xe2x88x92   Jacobi   xe2x88x92   Bellman systems for the postdefault case and the predefault case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies.

Timeconsistent equilibrium reinsurancexe2x80x93investment strategy for n competitive insurers under a new interaction mechanism and a general investment framework Abstract   We propose a new interaction mechanism for     n  xe2x89xa5  2     competitive insurers, under which we investigate the timeconsistent equilibrium reinsurancexe2x80x93investment strategy for    n    insurers. Each insurer can purchase reinsurance for reducing the claim risk and invest in the financial market for increasing his wealth. We consider a general investment framework which includes typical diversified and concentrated investment patterns in the literature. The objective of each insurer is to find a timeconsistent equilibrium reinsurancexe2x80x93investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By using the stochastic control technique, we obtain the explicit timeconsistent equilibrium reinsurancexe2x80x93investment strategies and corresponding equilibrium value functions. Through analyzing the general investment strategy, we find the criterion of whether the insurer should invest in a specific risky asset. Furthermore, we examine theoretically and numerically how the price parameters of risky assets affect the concrete investment pattern. We find that when the appreciation rate of one risky asset exceeds some times as those of other risky assets, the insurer will only invest in this asset; otherwise, the insurer will adopt the diversified investment pattern. We also numerically examine the influences of the number of insurers and the surplus jump on the timeconsistent equilibrium reinsurance strategy.