Abstract
Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asmussen S (1989) Risk theory in a Markovian environment. Scand Actuar J 1989(2):69–100
Asmussen S (2003) Applied probability and queues, 2nd edn. Springer, New York
Asmussen S, Albrecher H (2010) Ruin probabilities, 2nd edn. World Scientific, Singapore
Bäuerle N (1996) Some results about the expected ruin time in Markov-modulated risk models. Insur Math Econ 18:119–127
Brown M, Li S (2018) Sharp Bounds for Exponential Approximations of NWUE Distributions. Methodol Comput Appl Probab 20:875–896
Cao J, Roslan TRN, Zhang W (2018) Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching. Methodol Comput Appl Probab 20:1359–1379
Centeno ML (2002) Excess of loss reinsurance and Gerber’s inequality in the Sparre Anderson model. Insur Math Econ 31:415–427
Chen A, Delong Ł (2015) Optimal investment for a defined-contribution pension scheme under a regime-switching model. ASTIN Bull 45:397–419
Chen X, Xiao T, Yang X (2014) A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy. Insur Math Econ 54:76–83
Cheng J, Zhan Y (2020) Nonstationary l2-\(l_{\infty }\) filtering for Markov switching repeated scalar nonlinear systems with randomly occurring nonlinearities. Appl Math Comput 365(124714)
D’Amico G (2014) Moments analysis of a Markov-modulated risk model with stochastic interest rates. Commun Stoch Anal 8(2):227–246. https://doi.org/10.31390/cosa.8.2.06
Dębicka J (2013) An approach to the study of multistate insurance contracts. Appl Stoch Models Bus Ind 29(3):224–240
Feng R, Shimizu Y (2014) Potential measures for spectrally negative Markov additive processes with applications in ruin theory. Insur Math Econ 59:11–26
Gajek L, Rudź M (2013) Sharp approximations of ruin probabilities in the discrete time models. Scand Actuar J 2013(5):352–382
Gajek L, Rudź M (2017) A generalization of Gerber’s inequality for ruin probabilities in risk-switching models. Statist Probab Lett 129:236–240
Gajek L, Rudź M (2018a) Banach Contraction Principle and ruin probabilities in regime-switching models. Insur Math Econ 80:45–53
Gajek L, Rudź M (2018b) Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model. Methodol Comput Appl Probab Link at: https://doi.org/10.1007/s11009-018-9627-2
Gajek L, Rudź M (2018c) Deficit distributions at ruin in a regime-switching Sparre Andersen model. J Appl Anal 24:99–107
Gerber HU (1979) An introduction to mathematical risk theory. S. S Huebner Foundation for Insurance Education. University of Pennsylvania, Philadelphia
Gerber HU, Shiu ESW (1998) On the time value of ruin. N Amer Actuar J 2(1):48–72
Grandell J (1992) Aspects of risk theory, 2nd edn. Springer, New York
Guillou A, Loisel S, Stupfler G (2013) Estimation of the parameters of a Markov-modulated loss process in insurance. Insur Math Econ 53:388–404
Gzyl H, Mayoral S (2006) On a relationship between distorted and spectral risk measures. Faculty Working Papers 15/06, School of Economics and Business Administration, University of Navarra
Ignatieva K, Song A, Ziveyi J (2018) Fourier space time-stepping algorithm for valuing guaranteed minimum withdrawal benefits in variable annuities under regime-switching and stochastic mortality. ASTIN Bull 48:139–169
Jin Z, Liu G, Yang H (2020) Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for Markov regime-switching jump diffusion models. European J Oper Res 280:1130–1143
Kim B, Kim HS (2007) Moments of claims in a Markovian environment. Insur Math Econ 40:485–497
Landriault D, Li B, Li S (2015) Analysis of a drawdown-based regime-switching Lévy insurance model. Insur Math Econ 60:98–107
Lorek P (2017) Generalized gambler’s ruin problem: Explicit formulas via Siegmund duality. Methodol Comput Appl Probab 19:603–613
Liu Y, Privault N (2018) A recursive algorithm for selling at the ultimate maximum in regime-switching models. Methodol Comput Appl Probab 20:369–384
Lu Y (2006) On the severity of ruin in a Markov-modulated risk model. Scand Actuar J 2006(4):183–202
Rabehasaina L (2009) Risk processes with interest force in Markovian environment. Stoch Models 25(4):580–613
Reinhard JM (1984) On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. ASTIN Bull 14:23–43
Shen H, Wang Y, Xia J, Park JH, Wang Z (2019) Fault-tolerant leader-following consensus for multi-agent systems subject to semi-Markov switching topologies: An event-triggered control scheme. Nonlinear Anal Hybrid Syst 34:92–107
Silvestrov DS (2014) American-type options–stochastic approximation methods, vol. 1. De Gruyter Studies in Mathematics, 56. De Gruyter, Berlin
Silvestrov DS (2015) American-type options–stochastic approximation methods, vol. 2. De Gruyter Studies in Mathematics, 57. De Gruyter, Berlin
Wang G, Wang G, Yang H (2016) On a multi-dimensional risk model with regime switching. Insur Math Econ 68:73–83
Woo JK, Liu H (2018) Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model. Methodol Comput Appl Probab 20:1285–1318
Xu L, Zhang L, Yao D (2017) Optimal investment and reinsurance for an insurer under Markov-modulated financial market. Insur Math Econ 74:7–19
Acknowledgements
The authors thank the reviewers and the editors for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported by the National Science Centre, Poland (2014/13/B/HS4/03222).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gajek, L., Rudź, M. Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model. Methodol Comput Appl Probab 22, 1507–1528 (2020). https://doi.org/10.1007/s11009-020-09780-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-020-09780-3
Keywords
- Vector-valued risk operators
- Vector-valued loss measures at ruin
- Risk management based on internal models
- Markov chains
- NWUE
- Solvency II