×

zbMATH — the first resource for mathematics

Optimal dividends with incomplete information in the dual model. (English) Zbl 1189.91074
Summary: In [H. U. Gerber, E. S. W. Shiu and N. Smith, Insur. Math. Econ. 42, No. 1, 243–254 (2008; Zbl 1141.91513)], 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 [B. Avanzi, H. U. Gerber and E. S. W. Shiu, Insur. Math. Econ. 41, No. 1, 111–123 (2007; Zbl 1131.91026)]. 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.

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Avanzi, B.; Gerber, H.U.; Shiu, E.S., Optimal dividends in the dual model, Insurance: mathematics and economics, 41, 1, 111-123, (2007) · Zbl 1131.91026
[2] Badescu, A.L.; Stanford, D.A., A generalization of the de vylder approximation for the probability of ruin, Economic computation and economic cybernetics studies and research, 40, 3-4, 245-265, (2006)
[3] de Finetti, B., Su un’impostazione alternativa Della teoria collettiva del rischio, Transactions of the xvth international congress of actuaries, 2, 433-443, (1957)
[4] De Vylder, F., A practical solution to the problem of ultimate ruin probability, Scandinavian actuarial journal, 114-119, (1978)
[5] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[6] Gerber, H.U.; Shiu, E.S.W., Optimal dividends: analysis with Brownian motion, North American actuarial journal, 8, 1, 1-20, (2004) · Zbl 1085.62122
[7] Gerber, H.U.; Shiu, E.S.W.; Smith, N., Methods for estimating the optimal dividend barrier and the probability of ruin, Insurance: mathematics and economics, 42, 1, 243-254, (2008) · Zbl 1141.91513
[8] Grandell, J., Aspects of risk theory, (1991), Springer-Verlag New York · Zbl 0717.62100
[9] Klugman, S.A.; Panjer, H.H.; Willmot, G.E., Loss models: from data to decisions, (2004), John Wiley and Sons Hoboken, NJ · Zbl 1141.62343
[10] Miyasawa, K., An economic survival game, Journal of the operations research society of Japan, 4, 3, 95113, (1962)
[11] Seal, H.L., Stochastic theory of a risk business, (1969), John Wiley and Sons New York · Zbl 0196.23501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.