A stochastic differential reinsurance game

Abstract

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.

Optimal non-proportional reinsurance control and stochastic differential games

“…Our paper is an extension of Hipp and Taksar (2010) in the context of game theory. The stochastic differential games for a diffusion model with proportional reinsurance are studied in Zeng (2010b).…”

Stochastic Brownian Game of Absolute Dominance

“…Note that in the case of negative correlation the game is solvable. The solution can be found by a similar method to that used in [9] and so we omit it. Now we assume that ρ ≥ 0 and that V solves the FBI equations.…”

“…In [5], a two-player stochastic differential game was studied in a Lévy market, where Hamilton-JacobiBellman-Isaacs (HJBI) conditions were proved and the results applied to risk minimization problems. In [9], a proportional reinsurance game was formulated by maximizing or minimizing the exit probability of an interval. The value function and Nash equilibrium strategy were obtained explicitly.…”

“…We apply the modeling features of [1], [4], and [9], and formulate a competition between the two insurers using a two-player noncooperative differential game. The surplus processes are modeled by arithmetic Brownian motions with risk-free investment and Brownian perturbation.…”

A reinsurance game between two insurance companies with nonlinear risk processes

“…For maximizing/minimizing a payoff function depending on the difference of two insurance companies' surplus processes, Zeng (2010) and Taksar and Zeng (2011) studied a zero-sum stochastic differential game between two insurers by applying proportional and non-proportional reinsurance, respectively. Bensoussan et al (2014) studied the relative performance of two insurance companies under a nonzero sum stochastic differential game framework.…”