Hostname: page-component-5cf477f64f-qls9x Total loading time: 0 Render date: 2025-03-31T23:18:57.976Z Has data issue: false hasContentIssue false
  • English
  • Français

On the central management of risk networks

Part of: Special processes Stochastic processes Markov processes

Published online by Cambridge University Press:  17 March 2017

Florin Avram  and
Andreea Minca
Florin Avram*
Affiliation:
Université de Pau
Andreea Minca*
Affiliation:
Cornell University
*
* 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Multidimensional risk processSparre Andersen processMarkov additive processmatrix exponential approximationruin probability

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 221 - 237
Copyright
Copyright © Applied Probability Trust 2017 

References

[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.149–191.Google Scholar
[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. 37,203–233.Google Scholar
[3] Asmussen, S. (2003).Applied Probability and Queues(Appl. Math. 51).Springer,New York.Google Scholar
[4] Asmussen, S. and Albrecher, H. (2010).Ruin Probabilities(Adv. Ser. Statist. Sci. Appl. Prob. 14),2nd edn.World Scientific,Hackensack, NJ.Google Scholar
[5] Avanzi, B. (2009).Strategies for dividend distribution: a review.North Amer. Actuarial J. 13,217–251.Google Scholar
[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.Google Scholar
[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. 14,215–238.Google Scholar
[8] Avram, F.,Palmowski, Z. and Pistorius, M. R. (2007).On the optimal dividend problem for a spectrally negative Lévy process.Ann. Appl. Prob. 17,156–180.CrossRefGoogle Scholar
[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. 18,2421–2449.CrossRefGoogle Scholar
[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.Google Scholar
[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. 25,1868–1935.Google Scholar
[12] Azcue, P. and Muler, N. (2005).Optimal reinsurance and dividend distribution policies in the Cramér‒Lundberg model.Math. Finance 15,261–308.CrossRefGoogle Scholar
[13] Badescu, A. L.,Cheung, E. C. K. and Rabehasaina, L. (2011).A two-dimensional risk model with proportional reinsurance.J. Appl. Prob. 48,749–765.CrossRefGoogle Scholar
[14] Bertoin, J. (1998).Lévy Processes(Camb. Tracts Math. 121).Cambridge University Press.Google Scholar
[15] Chan, W.-S.,Yang, H. and Zhang, L. (2003).Some results on ruin probabilities in a two-dimensional risk model.Insurance Math. Econom. 32,345–358.Google Scholar
[16] Cuyt, A. et al. (2008).Handbook of Continued Fractions for Special Functions.Springer,New York.Google Scholar
[17] Czarna, I. and Renaud, J.-F. (2015).A note on Parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes.Preprint.Google Scholar
[18] Eisenberg, J. and Schmidli, H. (2011).Minimising expected discounted capital injections by reinsurance in a classical risk model.Scand. Actuarial J. 2011,155–176.Google Scholar
[19] Gerber, H. U. (1972).Games of economic survival with discrete-and continuous-income processes.Operat. Res. 20,37–45.CrossRefGoogle Scholar
[20] Gerber, H. U. (1984).Chains of reinsurance.Insurance Math. Econom. 3,43–48.CrossRefGoogle Scholar
[21] Ivanovs, J. (2011).One-sided Markov additive processes and related exit problems.Doctoral Thesis.Google Scholar
[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.Google Scholar
[23] Ivanovs, J. and Palmowski, Z. (2012).Occupation densities in solving exit problems for Markov additive processes and their reflections.Stoch. Process. Appl. 122,3342–3360.CrossRefGoogle Scholar
[24] Kulenko, N. and Schmidli, H. (2008).Optimal dividend strategies in a Cramér‒Lundberg model with capital injections.Insurance Math. Econom. 43,270–278.CrossRefGoogle Scholar
[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.97–186.Google Scholar
[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.121–135.Google Scholar
[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. 23,2357–2381.Google Scholar
[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. 18,1669–1680.Google Scholar
[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. 46,98–108.CrossRefGoogle Scholar
[30] Løkka, A. and Zervos, M. (2008).Optimal dividend and issuance of equity policies in the presence of proportional costs.Insurance Math. Econom. 42,954–961.CrossRefGoogle Scholar
[31] Neuts, M. (1969).The queue with Poisson input and general service times, treated as a branching process.Duke Math. J. 36,215–231.CrossRefGoogle Scholar
[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 22,55–75.CrossRefGoogle Scholar
[33] Von Dahlen, S. and Von Peter, G. (2012).Natural catastrophes and global reinsurance‒exploring the linkages.BIS Quart. Rev. 23–35.Google Scholar
[34] Yano, K. (2014).Functional limit theorems for processes pieced together from excursions.Preprint. Available at https://arXiv.org/abs/1309.2652v3.Google Scholar