Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T16:54:22.040Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE COMPOUND POISSON RISK MODEL WITH PERIODIC CAPITAL INJECTIONS

Published online by Cambridge University Press:  28 September 2017

Zhimin Zhang ,
Eric C.K. Cheung  and
Hailiang Yang
Zhimin Zhang
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China, E-Mail: [email protected]
Eric C.K. Cheung*
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, NSW 2052, Australia
Hailiang Yang
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, E-Mail: [email protected]
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The analysis of capital injection strategy in the literature of insurance risk models (e.g. Pafumi, 1998; Dickson and Waters, 2004) typically assumes that whenever the surplus becomes negative, the amount of shortfall is injected so that the company can continue its business forever. Recently, Nie et al. (2011) has proposed an alternative model in which capital is immediately injected to restore the surplus level to a positive level b when the surplus falls between zero and b, and the insurer is still subject to a positive ruin probability. Inspired by the idea of randomized observations in Albrecher et al. (2011b), in this paper, we further generalize Nie et al. (2011)'s model by assuming that capital injections are only allowed at a sequence of time points with inter-capital-injection times being Erlang distributed (so that deterministic time intervals can be approximated using the Erlangization technique in Asmussen et al. (2002)). When the claim amount is distributed as a combination of exponentials, explicit formulas for the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) and the expected total discounted cost of capital injections before ruin are obtained. The derivations rely on a resolvent density associated with an Erlang random variable, which is shown to admit an explicit expression that is of independent interest as well. We shall provide numerical examples, including an application in pricing a perpetual reinsurance contract that makes the capital injections and demonstration of how to minimize the ruin probability via reinsurance. Minimization of the expected discounted capital injections plus a penalty applied at ruin with respect to the frequency of injections and the critical level b will also be illustrated numerically.

Keywords

Compound Poisson risk modelperiodic capital injectionsGerber–Shiu expected discounted penalty functionresolvent measureperpetual reinsurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 435 - 477
Copyright
Copyright © Astin Bulletin 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecher, H., Bäuerle, N. and Thonhauser, S. (2011a) Optimal dividend-payout in random discrete time. Statistics and Risk Modeling, 28 (3), 251–276.CrossRefGoogle Scholar
Albrecher, H., Cheung, E.C.K. and Thonhauser, S. (2011b) Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bulletin, 41 (2), 645–672.Google Scholar
Albrecher, H., Cheung, E.C.K. and Thonhauser, S. (2013) Randomized observation periods for the compound Poisson risk model: The discounted penalty function. Scandinavian Actuarial Journal, 2013 (6), 424–452.Google Scholar
Albrecher, H. and Ivanovs, J. (2013) A risk model with an observer in a Markov environment. Risks, 1 (3), 148–161.CrossRefGoogle Scholar
Albrecher, H. and Ivanovs, J. (2017) Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Stochastic Processes and their Applications, 127 (2), 643–656.Google Scholar
Albrecher, H., Ivanovs, J. and Zhou, X. (2016) Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli, 22 (3), 1364–1382.Google Scholar
Asmussen, S., Avram, F. and Usabel, M. (2002) Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bulletin, 32 (2), 267–281.Google Scholar
Avanzi, B., Cheung, E.C.K., Wong, B. and Woo, J.-K. (2013) On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insurance: Mathematics and Economics, 52 (1), 98–113.Google Scholar
Avanzi, B., Tu, V. and Wong, B. (2014) On optimal periodic dividend strategies in the dual model with diffusion. Insurance: Mathematics and Economics, 55, 210–224.Google Scholar
Boxma, O.J., Jonsson, H., Resing, J.A.C. and Shneer, S. (2010) An alternating risk reserve process – Part II. Markov Processes And Related Fields, 16 (2), 425–446.Google Scholar
Cheung, E.C.K. (2010) Discussion of ‘A direct approach to the discounted penalty function’. North American Actuarial Journal, 14 (4), 441–445.CrossRefGoogle Scholar
Cheung, E.C.K. (2012) A unifying approach to the analysis of business with random gains. Scandinavian Actuarial Journal, 2012 (3), 153–182.CrossRefGoogle Scholar
Choi, M.C.H. and Cheung, E.C.K. (2014) On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions. Insurance: Mathematics and Economics, 59, 121–132.Google Scholar
Dickson, D.C.M. and Qazvini, M. (2016) Gerber–Shiu analysis of a risk model with capital injections. European Actuarial Journal, 6 (2), 409–440.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (2004) Some optimal dividends problems. ASTIN Bulletin, 34 (1), 49–74.Google Scholar
Dufresne, D. (2007) Fitting combinations of exponentials to probability distributions. Applied Stochastic Models in Business and Industry, 23 (1), 23–48.Google Scholar
Eisenberg, J. and Schmidli, H. (2011) Minimising expected discounted capital injections by reinsurance in a classical risk model. Scandinavian Actuarial Journal, 2011 (3), 155–176.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial Journal, 2 (1), 48–72.CrossRefGoogle Scholar
Kulenko, N. and Schmidli, H. (2008) Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insurance: Mathematics and Economics, 43 (2), 270–278.Google Scholar
Kyprianou, A.E. (2013) Gerber–Shiu Risk Theory. Cham, Heidelberg, New York, Dordrecht, London: Springer.Google Scholar
Landriault, D. and Willmot, G.E. (2008) On the Gerber–Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. Insurance: Mathematics and Economics, 42 (2), 600–608.Google Scholar
Liu, L. and Cheung, E.C.K. (2014) On a Gerber–Shiu type function and its applications in a dual semi-Markovian risk model. Applied Mathematics and Computation, 247, 1183–1201.CrossRefGoogle Scholar
Nie, C., Dickson, D.C.M. and Li, S. (2011) Minimizing the ruin probability through capital injections. Annals of Actuarial Science, 5 (2), 195–209.CrossRefGoogle Scholar
Nie, C., Dickson, D.C.M. and Li, S. (2015) The finite time ruin probability in a risk model with capital injections. Scandinavian Actuarial Journal, 2015 (4), 301–318.CrossRefGoogle Scholar
Pafumi, G. (1998) Discussion of ‘On the time value of ruin.’ North American Actuarial Journal, 2 (1), 75–76.Google Scholar
Ramaswami, V., Woolford, D.G. and Stanford, D.A. (2008) The Erlangization method for Markovian fluid flows. Annals of Operations Research, 160 (1), 215–225.Google Scholar
Stanford, D.A., Avram, F., Badescu, A.L., Breuer, L., Da Silva Soares, A. and Latouche, G. (2005) Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models. ASTIN Bulletin, 35 (1), 131–144.CrossRefGoogle Scholar
Stanford, D.A., Yu, K. and Ren, J. (2011) Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scandinavian Actuarial Journal, 2011 (1), 38–58.CrossRefGoogle Scholar
Zhang, Z. (2014) On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization, 10 (4), 1041–1058.Google Scholar
Zhang, Z. and Cheung, E.C.K. (2016) The Markov additive risk process under an Erlangized dividend barrier strategy. Methodology and Computing in Applied Probability, 18 (2), 275–306.Google Scholar
Zhang, Z., Cheung, E.C.K. and Yang, H. (2017) Lévy insurance risk process with Poissonian taxation. Scandinavian Actuarial Journal, 2017 (1), 51–87.Google Scholar