Premiums in a Dynamic Model of a Reinsurance Market

In this paper a continuous-time model of a reinsurance market is presented, which contains the principal components of uncertainty transparent in such a market: Uncertainty about the time instants at which accidents take place, and uncertainty about claim sizes given that accidents have occurred. Due to random jumps at random time points of the underlying claims processes, the absence of arbitrage opportunities is not sufficient to give unique premium functional in general. Market preferences are derived under a necessary condition for a general exchange equilibrium. Information constraints are found under which premiums of risks are determined. It is demonstrated how general reinsurance treaties can be uniquely split into proportional contracts and non-proportional ones. Several applications to reinsurance markets are given, and the results are compared to the corresponding theory of the classical one-period model of a reinsurance syndicate. This paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of partial equilibrium price formation at the heart of the economics of uncertainty. Key words: Reinsurance, intertemporal economic model, marked point processes, exchange equilibrium, no arbitrage, market’s marginal utility, martingales, random measures, nonlinear spanning, proportional and non-proportional treaties.