Optimal dividends with incomplete information in the dual model

In [Gerber, H.U., Shiu, E.S.W., Smith, N., 2008. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance: Math. Econ. 42 (1), 243-254], methods were analyzed for estimating the optimal dividend barrier (in the sense of de Finetti). In particular, De Vylder approximations and diffusion approximations are discussed. These methods are useful when only the first few moments of the claim amount distribution are known. The purpose of this paper is to examine these and other methods (such as the gamma approximations and the gamproc approximations) in the dual model, see [Avanzi, B., Gerber, H.U., Shiu, E.S., 2007. Optimal dividends in the dual model. Insurance: Math. Econ. 41 (1), 111-123]. The dual model is obtained if the roles of premiums and claims are exchanged. In other words, the company has random gains, which constitute a compound Poisson process, and expenses occur continuously at a constant rate. The approximations can easily be implemented, and their accuracy is surprisingly good. Several numerical illustrations enhance the paper.