Optimal Reinsurance and Dividend Distribution Policies in the Cramér‐lundberg Model

Abstract

We consider that the reserve of an insurance company follows a Cramér-Lundberg process. The management has the possibility of controlling the risk by means of reinsurance. Our aim is to find a dynamic choice of both the reinsurance policy and the dividend distribution strategy that maximizes the cumulative expected discounted dividend payouts. We study the usual cases of excess-of-loss and proportional reinsurance as well as the family of all possible reinsurance contracts. We characterize the optimal value function as the smallest viscosity solution of the associated Hamilton-JacobiBellman equation and we prove that there exists an optimal band strategy. We also describe the optimal value function for small initial reserves.

Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem

“…This result has been re-considered very recently in [3] for Cramér-Lundberg processes with a general jump distribution. In the latter paper it was shown that for an appropriate choice of jump distribution, the above described barrier strategy is not optimal.…”

Optimality results for dividend problems in insurance

“…This càglàd assumption is for instance used in Azcue & Muler [15] and Albrecher & Thonhauser [9] in a compound Poisson framework. Alternatively, it is also possible to consider previsible càdlàg strategies L, which preserve the càdlàg property of the risk process for the controlled process.…”

“…For obtaining explicit solutions and simple decision rules, one may want to focus on barrier or threshold strategies; for solving the problem in a general form one will want to deal with general càglàd cumulated dividend processes as specified in the previous section. The general problem for the classical Cramér-Lundberg risk reserve process was first solved by Gerber in [50] via a limit of an associated discrete problem and later on by means of stochastic control theory by Azcue & Muler [15], who also included a general reinsurance strategy as a second control possibility. See also Schmidli [111] and Mnif & Sulem [94] who allow for additional dynamic XL-reinsurance and Albrecher & Thonhauser [9] for a reserve process under a force of interest.…”

“…As already mentioned, he identified band strategies to be optimal in this context. Only recently Azcue & Muler [15] used the dynamic programming approach to obtain the HJB equation (19) for this problem (they also included a dynamic reinsurance possibility, see also Schmidli [110]). The two main difficulties which arise when looking at (19) are the question of differentiability and uniqueness of a solution.…”

“…Now it only remains to determine the dividend strategy associated to the correct solution V to (24). In the above example it turns out to be a band strategy defined by the sets In [110] an algorithmic procedure for obtaining V is described, in [15] and [9] explicit examples are constructed which demonstrate the necessity of an extended solution concept. A policy iteration algorithm for a related problem is constructed in [94].…”