scholarly journals A stochastic differential reinsurance game

2010 ◽  
Vol 47 (2) ◽  
pp. 335-349 ◽  
Author(s):  
Xudong Zeng
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Payoff Function ◽  
The Other ◽  
Insurance Companies ◽  
Stochastic Differential Game ◽  
Claim Size ◽  
Expected Payoff ◽  
Exit Probability ◽  
Risk Of Exposure

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

scholarly journals A stochastic differential reinsurance game

2010 ◽  
Vol 47 (02) ◽  
pp. 335-349 ◽  
Author(s):  
Xudong Zeng
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Payoff Function ◽  
The Other ◽  
Insurance Companies ◽  
Stochastic Differential Game ◽  
Claim Size ◽  
Expected Payoff ◽  
Exit Probability ◽  
Risk Of Exposure

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

scholarly journals Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

2020 ◽  
Vol 2020 ◽  
pp. 1-19 ◽  
Author(s):  
Danping Li ◽  
Ruiqing Chen ◽  
Cunfang Li
Keyword(s):  
Differential Game ◽  
Exponential Utility ◽  
Insurance Companies ◽  
Stochastic Differential Game ◽  
Investment Strategies ◽  
Small Company ◽  
Cev Model ◽  
Constant Elasticity ◽  
Optimal Reinsurance ◽  
Constant Elasticity Of Variance

This paper focuses on a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big company has sufficient asset to invest in the risky asset which is described by the constant elasticity of variance (CEV) model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. In this paper, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Furthermore, numerical examples are presented to show our results.

ON A CLASS OF NASH EQUILIBRIA WITH MEMORY STRATEGIES FOR NONZERO-SUM DIFFERENTIAL GAMES

2000 ◽  
Vol 02 (02n03) ◽  
pp. 173-192 ◽  
Author(s):  
JEAN MICHEL COULOMB ◽  
VLADIMIR GAITSGORY
Keyword(s):  
Nash Equilibrium ◽  
Feedback Control ◽  
Differential Games ◽  
Differential Game ◽  
Nash Equilibria ◽  
Payoff Function ◽  
Feedback Controls ◽  
Memory Strategies ◽  
Payoff Functions ◽  
With Memory

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

scholarly journals A Mixed Cooperative Dual to the Nash Equilibrium

Game Theory ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
H. W. Corley
Keyword(s):  
Nash Equilibrium ◽  
Mixed Strategy ◽  
The Other ◽  
Mixed Strategies ◽  
Mutual Support ◽  
Equilibrium Models ◽  
Expected Payoff ◽  
Berge Equilibrium

A mixed dual to the Nash equilibrium is defined for n-person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other n-1 players’ strategies. Conversely, in the dual equilibrium every n-1 players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each n>2 there exists a game for which no dual equilibrium exists.

A stochastic differential game for quadratic-linear diffusion processes

2016 ◽  
Vol 48 (4) ◽  
pp. 1161-1182 ◽  
Author(s):  
Shangzhen Luo
Keyword(s):  
Differential Game ◽  
Diffusion Processes ◽  
Risk Process ◽  
Equilibrium Strategy ◽  
Stochastic Differential Game ◽  
Linear Diffusion ◽  
Nash Equilibrium Strategy ◽  
Exit Probability ◽  
High Level ◽  
The Value Function

AbstractIn this paper we study a stochastic differential game between two insurers whose surplus processes are modelled by quadratic-linear diffusion processes. We consider an exit probability game. One insurer controls its risk process to minimize the probability that the surplus difference reaches a low level (indicating a disadvantaged surplus position of the insurer) before reaching a high level, while the other insurer aims to maximize the probability. We solve the game by finding the value function and the Nash equilibrium strategy in explicit forms.

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