Skip to main content


  • Zhimin Zhang (a1), Eric C.K. Cheung (a2) and Hailiang Yang (a3)

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.

Corresponding author
Hide All
Albrecher, H., Bäuerle, N. and Thonhauser, S. (2011a) Optimal dividend-payout in random discrete time. Statistics and Risk Modeling, 28 (3), 251276.
Albrecher, H., Cheung, E.C.K. and Thonhauser, S. (2011b) Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bulletin, 41 (2), 645672.
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), 424452.
Albrecher, H. and Ivanovs, J. (2013) A risk model with an observer in a Markov environment. Risks, 1 (3), 148161.
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), 643656.
Albrecher, H., Ivanovs, J. and Zhou, X. (2016) Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli, 22 (3), 13641382.
Asmussen, S., Avram, F. and Usabel, M. (2002) Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bulletin, 32 (2), 267281.
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), 98113.
Avanzi, B., Tu, V. and Wong, B. (2014) On optimal periodic dividend strategies in the dual model with diffusion. Insurance: Mathematics and Economics, 55, 210224.
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), 425446.
Cheung, E.C.K. (2010) Discussion of ‘A direct approach to the discounted penalty function’. North American Actuarial Journal, 14 (4), 441445.
Cheung, E.C.K. (2012) A unifying approach to the analysis of business with random gains. Scandinavian Actuarial Journal, 2012 (3), 153182.
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, 121132.
Dickson, D.C.M. and Qazvini, M. (2016) Gerber–Shiu analysis of a risk model with capital injections. European Actuarial Journal, 6 (2), 409440.
Dickson, D.C.M. and Waters, H.R. (2004) Some optimal dividends problems. ASTIN Bulletin, 34 (1), 4974.
Dufresne, D. (2007) Fitting combinations of exponentials to probability distributions. Applied Stochastic Models in Business and Industry, 23 (1), 2348.
Eisenberg, J. and Schmidli, H. (2011) Minimising expected discounted capital injections by reinsurance in a classical risk model. Scandinavian Actuarial Journal, 2011 (3), 155176.
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial Journal, 2 (1), 4872.
Kulenko, N. and Schmidli, H. (2008) Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insurance: Mathematics and Economics, 43 (2), 270278.
Kyprianou, A.E. (2013) Gerber–Shiu Risk Theory. Cham, Heidelberg, New York, Dordrecht, London: Springer.
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), 600608.
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, 11831201.
Nie, C., Dickson, D.C.M. and Li, S. (2011) Minimizing the ruin probability through capital injections. Annals of Actuarial Science, 5 (2), 195209.
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), 301318.
Pafumi, G. (1998) Discussion of ‘On the time value of ruin.’ North American Actuarial Journal, 2 (1), 7576.
Ramaswami, V., Woolford, D.G. and Stanford, D.A. (2008) The Erlangization method for Markovian fluid flows. Annals of Operations Research, 160 (1), 215225.
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), 131144.
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), 3858.
Zhang, Z. (2014) On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization, 10 (4), 10411058.
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), 275306.
Zhang, Z., Cheung, E.C.K. and Yang, H. (2017) Lévy insurance risk process with Poissonian taxation. Scandinavian Actuarial Journal, 2017 (1), 5187.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 1
Total number of PDF views: 116 *
Loading metrics...

Abstract views

Total abstract views: 427 *
Loading metrics...

* Views captured on Cambridge Core between 28th September 2017 - 21st July 2018. This data will be updated every 24 hours.