Doubly Stochastic Poisson Process and the Pricing of Catastrophe Reinsurance Contract

We use a doubly stochastic Poisson process (or the Cox process) to model the claim arrival process for catastrophic events. The shot noise process is used for the claim intensity function within the Cox process. The Cox process with shot noise intensity is examined by piecewise deterministic Markov process theory. We apply the Cox process incorporating the shot noise process as its intensity to price a stop-loss catastrophe reinsurance contract. The asymptotic (stationary) distribution of the claim intensity is used to derive pricing formulae for a stop-loss reinsurance contract for catastrophic events. We achieve an absence of arbitrage opportunities in the market by using an equivalent martingale probability measure in the pricing model for catastrophe reinsurance contract. The Esscher transform is employed to change the probability measure.

KEYWORDS: Doubly stochastic Poisson process, Shot noise process, Piecewise deterministic Markov process theory, Stop-loss reinsurance contract, Equivalent martingale probability measure, Esscher transform.