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On the central management of risk networks

  • Florin Avram (a1) and Andreea Minca (a2)

Abstract

In this paper we identify three questions concerning the management of risk networks with a central branch, which may be solved using the extensive machinery available for one-dimensional risk models. First, we propose a criterion for judging whether a subsidiary is viable by its readiness to pay dividends to the central branch, as reflected by the optimality of the zero-level dividend barrier. Next, for a deterministic central branch which must bailout a single subsidiary each time its surplus becomes negative, we determine the optimal bailout policy, as well as the ruin probability and other risk measures, in closed form. Moreover, we extend these results to the case of hierarchical networks. Finally, for nondeterministic central branches with one subsidiary, we compute approximate risk measures by applying rational approximations, and by using the recently developed matrix scale methodology.

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Copyright

Corresponding author

* Postal address: Département de Mathématiques, Université de Pau, Pau 64000, France. Email address: florin.avram@univ-pau.fr
** Postal address: School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14850, USA. Email address: acm299@cornell.edu

References

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[1] Abate, J. and Whitt, W. (1988).Approximations for the M/M/1 busy-period distribution.In Queueing Theory and Its Applications(CWI Monogr. 7),North-Holland,Amsterdam,pp.149191.
[2] Albrecher, H.,Hartinger, J. and Thonhauser, S. (2007).On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model.ASTIN Bull. 37,203233.
[3] Asmussen, S. (2003).Applied Probability and Queues(Appl. Math. 51).Springer,New York.
[4] Asmussen, S. and Albrecher, H. (2010).Ruin Probabilities(Adv. Ser. Statist. Sci. Appl. Prob. 14),2nd edn.World Scientific,Hackensack, NJ.
[5] Avanzi, B. (2009).Strategies for dividend distribution: a review.North Amer. Actuarial J. 13,217251.
[6] Avram, F. and Minca, A. (2015).Steps towards a management toolkit for central branch risk networks, using rational approximations and matrix scale functions.In Modern Trends in Controlled Stochastic Processes: Theory and Applications,Luniver Press,Bristol.
[7] Avram, F.,Kyprianou, A. E. and Pistorius, M. R. (2004).Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options.Ann. Appl. Prob. 14,215238.
[8] Avram, F.,Palmowski, Z. and Pistorius, M. R. (2007).On the optimal dividend problem for a spectrally negative Lévy process.Ann. Appl. Prob. 17,156180.
[9] Avram, F.,Palmowski, Z. and Pistorius, M. R. (2008).Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results.Ann. Appl. Prob. 18,24212449.
[10] Avram, F.,Palmowski, Z. and Pistorius, M. (2009).On optimal dividend distribution for a Cramér‒Lundberg process with exponential jumps in the presence of a linear Gerber-Shiu penalty function.Monogr. Semin. Mat. García Galdeano nn 1,10.
[11] Avram, F.,Palmowski, Z. and Pistorius, M. R. (2015).On Gerber‒Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function.Ann. Appl. Prob. 25,18681935.
[12] Azcue, P. and Muler, N. (2005).Optimal reinsurance and dividend distribution policies in the Cramér‒Lundberg model.Math. Finance 15,261308.
[13] Badescu, A. L.,Cheung, E. C. K. and Rabehasaina, L. (2011).A two-dimensional risk model with proportional reinsurance.J. Appl. Prob. 48,749765.
[14] Bertoin, J. (1998).Lévy Processes(Camb. Tracts Math. 121).Cambridge University Press.
[15] Chan, W.-S.,Yang, H. and Zhang, L. (2003).Some results on ruin probabilities in a two-dimensional risk model.Insurance Math. Econom. 32,345358.
[16] Cuyt, A. et al. (2008).Handbook of Continued Fractions for Special Functions.Springer,New York.
[17] Czarna, I. and Renaud, J.-F. (2015).A note on Parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes.Preprint.
[18] Eisenberg, J. and Schmidli, H. (2011).Minimising expected discounted capital injections by reinsurance in a classical risk model.Scand. Actuarial J. 2011,155176.
[19] Gerber, H. U. (1972).Games of economic survival with discrete-and continuous-income processes.Operat. Res. 20,3745.
[20] Gerber, H. U. (1984).Chains of reinsurance.Insurance Math. Econom. 3,4348.
[21] Ivanovs, J. (2011).One-sided Markov additive processes and related exit problems.Doctoral Thesis.
[22] Ivanovs, J. (2013).Spectrally-negative Markov additive processes: in continuous time, version 1.0.Mathematica 8 package. Available at https://sites.google.com/site/jevgenijsivanovs/files.
[23] Ivanovs, J. and Palmowski, Z. (2012).Occupation densities in solving exit problems for Markov additive processes and their reflections.Stoch. Process. Appl. 122,33423360.
[24] Kulenko, N. and Schmidli, H. (2008).Optimal dividend strategies in a Cramér‒Lundberg model with capital injections.Insurance Math. Econom. 43,270278.
[25] Kuznetsov, A.,Kyprianou, A. E. and Rivero, V. (2013).The theory of scale functions for spectrally negative Lévy processes.In Lévy Matters II(Lecture Notes Math. 2061),Springer,Berlin,pp.97186.
[26] Kyprianou, A. E. and Palmowski, Z. (2008).Fluctuations of spectrally negative Markov additive processes.In Séminaire de Probabilités XLI(Lecture Notes Math. 1934),Springer,Berlin,pp.121135.
[27] Lambert, A.,Simatos, F. and Zwart, B. (2013).Scaling limits via excursion theory: interplay between Crump‒Mode‒Jagers branching processes and processor-sharing queues.Ann. Appl. Prob. 23,23572381.
[28] Loeffen, R. L. (2008).On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes.Ann. Appl. Prob. 18,16691680.
[29] Loeffen, R. L. and Renaud, J.-F. (2010).De Finetti's optimal dividends problem with an affine penalty function at ruin.Insurance Math. Econom. 46,98108.
[30] Løkka, A. and Zervos, M. (2008).Optimal dividend and issuance of equity policies in the presence of proportional costs.Insurance Math. Econom. 42,954961.
[31] Neuts, M. (1969).The queue with Poisson input and general service times, treated as a branching process.Duke Math. J. 36,215231.
[32] 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 22,5575.
[33] Von Dahlen, S. and Von Peter, G. (2012).Natural catastrophes and global reinsurance‒exploring the linkages.BIS Quart. Rev. 2335.
[34] Yano, K. (2014).Functional limit theorems for processes pieced together from excursions.Preprint. Available at https://arXiv.org/abs/1309.2652v3.

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On the central management of risk networks

  • Florin Avram (a1) and Andreea Minca (a2)

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