A Stackelberg reinsurance–investment game with asymmetric information and delay

ABSTRACT

This paper investigates a Stackelberg stochastic differential reinsurance–investment game problem, in which the reinsurer is the leader and the insurer is the follower. The unequal status of the reinsurer and the insurer in the financial market is characterized by the asymmetric information model. We consider their wealth processes with delay to characterize bounded memory. The objective of the reinsurer is to find the optimal premium pricing strategy and investment strategy such that its constant absolute risk aversion (CARA) utility of the combination of terminal wealth and average performance wealth is maximized. The objective of the insurer is to find the optimal reinsurance strategy and investment strategy such that its CARA utility of the relative performance is maximized. We derive the equilibrium strategy explicitly for the game by solving corresponding Hamilton–Jacobi–Bellman equations sequentially. Then, we establish the verification theorem. The equilibrium investment strategy indicates that the insurer with less information completely imitates the investment strategy of the reinsurer who has more information on the financial market. Further, we find that the effect of the delay weight on the equilibrium strategy is related to the length of delay time. Finally, we present some numerical examples to demonstrate the findings.

KEYWORDS:

• Stackelberg stochastic differential game
• reinsurance
• investment
• asymmetric information
• delay

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 91B30
• 91A23
• 90B50
• 62P05
• 91G80

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 From $A_F=r_0-B_F-C_F$, $C_F={\eta }_F{e}^{-{\alpha }_F{h}_F}$ and $B_F{e}^{-{\alpha }_F{h}_F}=({\alpha }_F+A_F+{\eta }_F)C_F$, we can get that $A_F=\frac{1}{1+{\eta }_F}[r_0-({\alpha }_F+{\eta }_F){\eta }_F-{\eta }_F{e}^{-{\alpha }_F{h}_F}]$. From $A_L=r_0-B_L-C_L$, $C_L={\eta }_L{e}^{-{\alpha }_L{h}_L}$, $B_L{e}^{-{\alpha }_L{h}_L}=({\alpha }_L+A_L+{\eta }_L)C_L$ and the $A_F+{\eta }_F=A_L+{\eta }_L$, we can get that ${\eta }_L=\frac{(r_0-1+e^{-{\alpha }_F{h}_F}+{\alpha }_F){\eta }_F}{(r_0-1+e^{-{\alpha }_L{h}_L}+{\alpha }_L)+({\alpha }_L-{\alpha }_F+e^{-{\alpha }_L{h}_L}-e^{-{\alpha }_F{h}_F}){\eta }_F}$. Using the same method, we know that ${\eta }_F=\frac{(r_0-1+e^{-{\alpha }_L{h}_L}+{\alpha }_L){\eta }_L}{(r_0-1+e^{-{\alpha }_F{h}_F}+{\alpha }_F)+({\alpha }_F-{\alpha }_L+e^{-{\alpha }_F{h}_F}-e^{-{\alpha }_L{h}_L}){\eta }_L}$.

2 Firstly, the time interval $[0,T]$ is equally divided into 1001 time points (i.e. $0,0.01,0.02,\dots ,10$). Then the stock price process in CEV model under P measure is simulated by using Monte Carlo method. Finally, Figure 1 can be drawn according to the expression of the optimal strategies.

3 Note: if $N^{{\theta }_F}\left(t\right)\ge 1$ is satisfied, i.e. case (1) in Theorem 3.1 occurs, then $p^{\ast }\left(t\right)$ can be any value in the interval $[c_F,\overline{c}]$, so we set $p^{\ast }\left(t\right)=\overline{c}$ in this case.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (Nos. 71771082, 71801091) and Hunan Provincial Natural Science Foundation of China (No. 2017JJ1012).