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An optimal reinsurance problem in the Cramér-Lundberg model. (English) Zbl 1377.93174
Summary: In this article, we consider the surplus process of an insurance company within the Cramér-Lundberg framework with the intention of controlling its performance by means of dynamic reinsurance. Our aim is to find a general dynamic reinsurance strategy that maximizes the expected discounted surplus level integrated over time. Using analytical methods we identify the value function as a particular solution to the associated Hamilton-Jacobi-Bellman equation. This approach leads to an implementable numerical method for approximating the value function and optimal reinsurance strategy. Furthermore we give some examples illustrating the applicability of this method for proportional and XL-reinsurance treaties.

MSC:
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)
93A30 Mathematical modelling of systems (MSC2010)
60G99 Stochastic processes
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[1] Asmussen S, Albrecher H (2010) Ruin probabilities, 2nd edn. World Scientific, Singapore · Zbl 1247.91080
[2] Azcue, P; Muler, N, Optimal reinsurance and dividend distribution policies in the cramér-lundberg model, Math Finance, 15, 261-308, (2005) · Zbl 1136.91016
[3] Azcue P, Muler N (2014) Stochastic optimization in insurance: a dynamic programming approach. Springer, New York · Zbl 1308.91004
[4] Bäuerle N, Rieder U (2011) Markov decision processes with applications to finance. Springer, New York · Zbl 1236.90004
[5] Borch KH (1974) The mathematical theory of Insurance: an annotated selection of papers on insurance published 1960-1972. Lexington Books, Lexington
[6] Cai, J; Feng, R; Willmot, GE, On the expectation of total discounted operating costs up to default and its applications, Adv Appl Probab, 41, 495-522, (2009) · Zbl 1173.91023
[7] Centeno, L, Measuring the effects of reinsurance by the adjustment coefficient, Insur Math Econ, 5, 169-182, (1986) · Zbl 0598.62141
[8] de Centeno ML (2002) Measuring the effects of reinsurance by the adjustment coefficient in the sparre anderson model. Insur Math Econ 30(1):37-49 · Zbl 1037.62106
[9] Eisenberg, J, On optimal control of capital injections by reinsurance and investments, Blätter der Dtsch Ges für Versicher- und Finanz, 31, 329-345, (2010) · Zbl 1205.91080
[10] Guerra M, Centeno ML (2008) Optimal reinsurance policy: the adjustment coefficient and the expected utility criteria. Insur Math Econ 42(2):529-539 · Zbl 1152.91583
[11] Hipp, C; Taksar, M, Optimal non-proportional reinsurance control, Insur Math Econ, 47, 246-254, (2010) · Zbl 1231.91199
[12] Hipp, C; Vogt, M, Optimal dynamic XL reinsurance, ASTIN Bull, 33, 193-207, (2003) · Zbl 1059.93135
[13] Højgaard, B; Taksar, M, Optimal proportional reinsurance policies for diffusion models, Scand Actuar J, 2, 166-180, (1998) · Zbl 1075.91559
[14] Højgaard, B; Taksar, M, Optimal proportional reinsurance policies for diffusion models with transaction costs, Insur Math Econ, 22, 41-51, (1998) · Zbl 1093.91518
[15] Højgaard, B; Taksar, M, Controlling risk exposure and dividends payout schemes: insurance company example, Math Finance, 9, 153-182, (1999) · Zbl 0999.91052
[16] Irgens, C; Paulsen, J, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insur Math Econ, 35, 21-51, (2004) · Zbl 1052.62107
[17] Kenneth J (1973) Arrow. Optimal insurance and generalized deductibles. Rand, Santa Monica
[18] Korn, R; Menkens, O; Steffensen, M, Worst-case-optimal dynamic reinsurance for large claims, Eur Actuar J, 2, 21-48, (2012) · Zbl 1269.91044
[19] Mnif, M; Sulem, A, Optimal risk control and dividend policies under excess of loss reinsurance, Stochastics, 77, 455-476, (2005) · Zbl 1076.93046
[20] Raviv, A, The design of an optimal insurance policy, Am Econ Rev, 69, 84-96, (1979)
[21] Rogers LCG, Williams D (1994) Diffusions, Markov processes, and martingales, vol 1. Wiley, Hoboken · Zbl 0826.60002
[22] Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, Hoboken · Zbl 0940.60005
[23] Schäl M (1998) On piecewise deterministic Markov control processes: control of jumps and of risk processes in insurance. Insur Math Econ 22(1):75-91. The interplay between insurance, finance and control (Aarhus, 1997)
[24] Schäl, M, On discrete-time dynamic programming in insurance: exponential utility and minimizing the ruin probability, Scand Actuar J, 3, 189-210, (2004) · Zbl 1141.91031
[25] Schmidli, H, Optimal proportional reinsurance policies in a dynamic setting, Scand Actuar J, 1, 55-68, (2001) · Zbl 0971.91039
[26] Schmidli H (2008) Stochastic control in insurance. Springer, New York · Zbl 1133.93002
[27] Schmidli, H; Hald, M, On the maximisation of the adjustment coefficient under proportional reinsurance, ASTIN Bull, 34, 75-83, (2004) · Zbl 1095.91033
[28] Taksar, M, Optimal risk and dividend distribution control models for an insurance company, Math Methods Oper Res, 51, 1-42, (2000) · Zbl 0947.91043
[29] Waters, HR, Some mathematical aspects of reinsurance, Insur Math Econ, 2, 17-26, (1983) · Zbl 0505.62085
[30] Wheeden RL, Zygmund A (1977) Measure and Integral: an introduction to real analysis. Marcel Dekker Inc., New York · Zbl 0362.26004
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