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Minimising expected discounted capital injections by reinsurance in a classical risk model. (English) Zbl 1277.60145
Summary: In this paper we consider a classical continuous-time risk model, where the claims are reinsured by some reinsurance with retention level \(b\in[0,\tilde b]\), where \(b=\tilde b\) means ‘no reinsurance’ and \(b=0\) means ‘full reinsurance’. The insurer can change the retention level continuously. To prevent negative surplus the insurer has to inject additional capital. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We show that an optimal reinsurance strategy exists. For some special cases we will be able to give the optimal strategy explicitly. In other cases the method will be illustrated only numerically.

MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
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[1] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[2] Bremaud P., Point processes and queues, martingale dynamics (1981) · Zbl 0478.60004 · doi:10.1007/978-1-4684-9477-8
[3] de Finetti B., Transactions of the XVth International Congress of Actuaries 2 pp 433– (1957)
[4] DOI: 10.2143/AST.34.1.504954 · Zbl 1097.91040 · doi:10.2143/AST.34.1.504954
[5] Eisenberg J., Optimal control of capital injections by reinsurance in a diffusion approximation. Preprint (2008)
[6] Gerber H. U., Schweizerische Vereinigung der Versicherungsmathematiker Mitteilungen 69 pp 185– (1969)
[7] Gerber H. U., Insurance: Mathematics and Economics 69 pp 145– (1997)
[8] Grandell J., Aspects of risk theory (1991) · Zbl 0717.62100
[9] DOI: 10.1016/j.insmatheco.2008.05.013 · Zbl 1189.91075 · doi:10.1016/j.insmatheco.2008.05.013
[10] DOI: 10.1239/aap/1183667616 · Zbl 1122.60076 · doi:10.1239/aap/1183667616
[11] DOI: 10.1002/9780470317044 · doi:10.1002/9780470317044
[12] Schmidli H., Stochastic control in insurance (2008) · Zbl 1133.93002
[13] Schmidli H., On the Gerber–Shiu function and change of measure. Preprint (2010) · Zbl 1231.91232
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