In the insurance market, the insurers who provide catastrophe insurance face with the risk of rare, but huge catastrophe claims. The introduction of catastrophe related securities into the market place provide the insurers the instruments to hedge some of the catastrophe risks they are facing. A catastrophe related security is tied to the prespecified catastrophe claims, while independent to other claims. It is the ties between the catastrophe related securities and the catastrophe insurance claims that make the catastrophe related securities special to the insurers.
The catastrophe securities and the insurance claims are not just correlated each other but simply bounded together for those claims specified in the catastrophe securities. However, the insurers cannot trade away the risk of those claims that are not covered by the catastrophe securities, because the catastrophe related securities do not cover every possible claims. In other words, the catastrophe insurance market is incomplete. Therefore, this partial tie characterizes the relation between the catastrophe securities and the claims. The partial tie poses a different optimal investment problem, what is the best policy to participate in the catastrophe security market, for a catastrophe insurer.
In this paper we study the optimal investment policy for the insurers in the business of catastrophe insurance. The insurer investment problems have been investigated by Browne (1994), Frolova, Kabanov, and Pergamenshchikov (2002), Gaier and Grandits (2002), and Gaier, Grandits, and Schachermayer (2003) in the special case of a geometric Brownian motion as an investment process. In the case of compound Poisson as insurance claim process and the geometric Brownian motion as the investment process the problem is investigated by Hipp and Plum (2000), Kalashnikov and Norberg (2002), and Paulsen (2002).
Emmer and Klüppelberg (2004) investigate the case where both insurance claims and the market security follow Lévy process. In those studies the investment securities and the insurance claims are assumed either independent or correlated, while statistic correlation is inadequate to model the relation between the catastrophe related securities and the catastrophe insurance claims. In this study, we model this partially tied relation between the catastrophe securities and the catastrophe claims. The catastrophe claims are rare events. Brownian motion is not a good model for rare events. In this study, both the catastrophe claims and the catastrophe related securities are driven by compound Poisson process.
