Source: J. Appl. Probab. Volume 47, Number 2
(2010), 335-349.
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.
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