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Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

Part of: Markov processes Mathematical economics Special processes

Published online by Cambridge University Press:  14 July 2016

Julia Eisenberg  and
Hanspeter Schmidli
Julia Eisenberg*
Affiliation:
University of Cologne
Hanspeter Schmidli*
Affiliation:
University of Cologne
*
∗ Current address: Financial and Actuarial Mathematics, Vienna University of Technology, A-1040 Vienna, Austria.
∗∗ Postal address: Institute of Mathematics, University of Cologne, Weyertal 86-90, D-50931 Cologne, Germany. Email address: [email protected]
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Abstract

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We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t }t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

Keywords

Optimal controlstochastic controlHamilton-Jacobi-Bellman equationcapital injectionclassical risk modelconstant interest rateriskless asset

MSC classification

Secondary: 60K10: Applications (reliability, demand theory, etc.) 91B30: Risk theory, insurance 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 733 - 748
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
[2] Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261–308.Google Scholar
[3] Eisenberg, J. and Schmidli, H. (2009). Optimal control of capital injections by reinsurance in a diffusion approximation. Blätter DGVFM 30, 1–13.Google Scholar
[4] Eisenberg, J. and Schmidli, H. (2011). Minimising expected discounted capital injections by reinsurance in a classical risk model. To appear in Scand. Actuarial J. Google Scholar
[5] Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.Google Scholar
[6] Hipp, C. and Plum, M. (2000). Optimal investment for insurers. Insurance Math. Econom. 27, 215–228.Google Scholar
[7] Hipp, C. and Schmidli, H. (2004). Asymptotics of ruin probabilities for controlled risk processes in the small claims case. Scand. Actuarial J. 2004, 321–335.Google Scholar
[8] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2. Cambridge University Press.Google Scholar
[9] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
[10] Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuarial J. 2001, 55–68.Google Scholar
[11] Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Prob. 12, 890–907.Google Scholar
[12] Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.Google Scholar
[13] Schmidli, H. (2010). On the Gerber–Shiu function and change of measure. Insurance Math. Econom. 46, 3–11.Google Scholar
[14] Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optimization 22, 55–75.Google Scholar