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On fair reinsurance premiums; capital injections in a perturbed risk model. (English) Zbl 1416.91157

Summary: We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company’s survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of M. Huzak et al. [Ann. Appl. Probab. 14, No. 3, 1378–1397 (2004; Zbl 1061.60075)] and the first author [Eur. Actuar. J. 4, No. 1, 219–246 (2014; Zbl 1307.91094)] on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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[1] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific Publishing London · Zbl 1247.91080
[2] Avram, F.; Loke, S.-H., On central branch/reinsurance risk networks: exact results and heuristics, Risks, 6, 35, (2018)
[3] Avram, F.; Palmowski, Z.; Pistorius, M. R., On the optimal dividend problem for a spectrally negative Lévy process, Ann. Appl. Probab., 17, 1, 156-180, (2007) · Zbl 1136.60032
[4] Ben Salah, Z., On a generalization of the expected discounted penalty function to include deficits at and beyond ruin, Eur. Actuar. J., 4, 4, 219-246, (2014) · Zbl 1307.91094
[5] Ben Salah, Z.; Guérin, H.; Morales, M.; Omidi Firouzi, H., On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory, Eur. Actuar. J., 5, 2, 381-425, (2015) · Zbl 1396.91292
[6] Bertoin, J., Lévy processes, (1998), Cambridge University Press · Zbl 0938.60005
[7] Biffis, E.; Kyprianou, A. E., A note on scale function and the time value of ruin, Insurance Math. Econom., 46, 85-91, (2010) · Zbl 1231.91145
[8] Biffis, E.; Morales, M., On a generalization of the gerber-shiu function to path-dependent penalties, Insurance Math. Econom., 46, 92-97, (2010) · Zbl 1231.91146
[9] Centeno, M. L.; Simões, O., Optimal reinsurance, Rev. R. Acad. Cienc. A, Mat., 103, 2, 387-405, (2009) · Zbl 1181.91090
[10] Doney, R.; Kyprianou, A. E., Overshoots and undershoots of Lévy processes, Ann. Appl. Probab., 16, 1, 91-106, (2006) · Zbl 1101.60029
[11] Dufresne, F.; Gerber, H., Risk theory for compound Poisson process that is perturbed by diffusion, Insurance Math. Econom., 10, 51-59, (1991) · Zbl 0723.62065
[12] Dufresne, F.; Gerber, H.; Shiu, E. W., Risk theory with gamma process, Astin Bull., 21, 177-192, (1991)
[13] Einsenberg, J.; Schmidli, H., Minimising expected discounted capital injections by reinsurance in a classical risk model, Scand. Actuar. J., 3, 155-176, (2011) · Zbl 1277.60145
[14] Furrer, H. J., Risk processes perturbed by \(\alpha\)-stable Lévy motion, Scand. Actuar. J., 1, 59-74, (1998) · Zbl 1026.60516
[15] Furrer, H. J.; Schmidli, H., Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion, Insurance Math. Econom., 15, 23-36, (1994) · Zbl 0814.62066
[16] Garrido, J.; Morales, M., On the expected penalty function for Lévy risk process, N. Am. Actuar. J., 10, 4, 196-218, (2006)
[17] Gerber, H.; Landry, B., On a discounted penalty at ruin in a jump-diffusion and perpetual put option, Insurance Math. Econom., 22, 263-276, (1998) · Zbl 0924.60075
[18] Gerber, H.; Shiu, E., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance Math. Econom., 21, 129-137, (1997) · Zbl 0894.90047
[19] Gerber, H.; Shiu, E., On the time value of ruin, N. Am. Actuar. J., 2, 1, 48-78, (1998) · Zbl 1081.60550
[20] Gerber, H.; Shiu, E., Pricing perpetual options for jump processes, N. Am. Actuar. J., 2, 3, 101-112, (1998) · Zbl 1081.91528
[21] Huzak, M.; Perman, M.; Sikic, H.; Vondracek, Z., Ruin probabilities and decompositions for general perturbed risk processes, Ann. Appl. Probab., 14, 3, 1378-1397, (2004) · Zbl 1061.60075
[22] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., The theory of scale functions for spectrally negative Lévy processes, (Lévy Matters II, Lecture Notes in Mathematics, vol. 2061, (2013), Springer), 97-186 · Zbl 1261.60047
[23] Kyprianou, A. E., Introductory lectures on Lévy processes with applications, (2006), Universitext, Springer · Zbl 1104.60001
[24] Lundberg, F., 1903. Approximerad framstallning av sannolikhetsfunktionen. aterforsakring av kollektivrisker. Akad Afhandling Almqvist och Wiksell, Uppsala.; Lundberg, F., 1903. Approximerad framstallning av sannolikhetsfunktionen. aterforsakring av kollektivrisker. Akad Afhandling Almqvist och Wiksell, Uppsala.
[25] Yang, H.; Zhang, L., Spectrally negative Lévy processes with applications in risk theory, Adv. Appl. Probab., 33, 281-291, (2001) · Zbl 0978.60104
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