Sensitivity analysis of some applied probability models

E. V. Bulinskaya ; B. I. Shigida

Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 3, pp. 19-35.

Résumé

During the last two decades, new models were developed in actuarial sciences. Different notions of insurance company ruin (bankruptcy) and other objective functions evaluating the company performance were introduced. Several types of decision (such as dividend payment, reinsurance, investment) are used for optimization of company functioning. Therefore, it is necessary to be sure that the model under consideration is stable with respect to parameter fluctuation and perturbation of underlying stochastic processes. The aim of this paper is the description of methods for investigation of these problems and presentation of recent results concerning some insurance models. Numerical results are also included.

Bibliographie

[1] Abdallah A., Boucher J. P., Cossette H., “Modeling dependence between loss triangles with hierarchical Archimedean copulas”, ASTIN Bull., 45 (2015), 577–599 | DOI | MR | Zbl

[2] Albrecher H., Gerber H. U., Shiu E. S. W., “The optimal dividend barrier in the Gamma-Omega model”, Europ. Actuar. J., 1 (2011), 43–55 | DOI | MR | Zbl

[3] D'Arcy S. P., “On becoming an actuary of the fourth kind”, Proc. Casualty Actuar. Soc., 177 (2005), 745–754

[4] Asmussen S., Albrecher H., Ruin Probabilities, World Scientific, New Jersey, 2010 | MR | Zbl

[5] Avanzi B., “Strategies for dividend distribution: A review”, North Am. Actuar. J., 13:2 (2009), 217–251 | DOI | MR

[6] Avanzi B., Gerber H. U., Shiu E. S. W., “Optimal dividends in the dual model”, Insurance Math. Econ., 41:1 (2007), 111–123 | DOI | MR | Zbl

[7] Bateman H., Table of Integral Transforms, v. I, McGraw-Hill, New York, 1954 | MR

[8] Bayraktar E., Egami M., “Optimizing venture capital investment in a jump diffusion model”, Math. Methods Operat. Res., 67:1 (2008), 21–42 | DOI | MR | Zbl

[9] Breuer L., Badescu A., “A generalised Gerber–Shiu measure for Markov-additive risk processes with phase-type claims and capital injections”, Scand. Actuar. J., 2014, no. 2, 93–115 | DOI | MR | Zbl

[10] Bulinskaya E. V., “Systems stability and optimal control”, J. Math. Sci., 92:3 (1998), 3857–3872 | DOI | MR | Zbl

[11] Bulinskaya E. V., “On a cost approach in insurance”, Rev. Appl. Ind. Math., 10:2 (2003), 376–386

[12] Bulinskaya E. V., “Sensitivity analysis of some applied models”, Pliska Stud. Math. Bulgar., 18 (2007), 57–90 | MR

[13] Bulinskaya E. V., “Stochastic insurance models: their optimality and stability”, Advances in Data Analysis, ed. Skiadas Ch. H., Birkhäuser, Boston, 2010, 129–140 | DOI | MR

[14] Bulinskaya E., “New research directions in modern actuarial sciences Modern Problems of Stochastic Analysis and Statistics”, Selected Contributions in Honor of Valentin Konakov, Springer Ser. Math. Stat., ed. Panov V., Springer, Berlin, 2017 | MR

[15] Bulinskaya E., “Stability problems in modern actuarial sciences”, Proc. of the Int. Conf. ACMPT-2017, 23–26 October 2017, Moscow, 2017, 435–439

[16] Bulinskaya E., Gusak J., “Optimal control and sensitivity analysis for two risk models”, Commun. Statistics Simulat. Comput., 45:5 (2016), 1451–1466 | DOI | MR | Zbl

[17] Cruz M. G., Peters G. W., Shevchenko P. V., Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, Wiley, 2015 | MR

[18] Czarna I., Palmowski Z., “Ruin probability with Parisian delay for a spectrally negative Lévy risk process”, J. Appl. Probab., 48 (2011), 984–1002 | DOI | MR | Zbl

[19] Dassios A., Wu Sh., Parisian ruin with exponential claims, stats.lse.ac.uk/angelos/docs/exponentialjump.pdf

[20] De Finetti B., “Su un'impostazione alternativa della teoria collettiva del rischio”, Transactions of the XV-th Int. Congress of Actuaries, v. 2, 1957, 433–443

[21] Fu D., Guo Y., “On the compound Poisson model with debit interest under absolute ruin”, Int. J. Sci. Res., 5:6 (2016), 1872–1875

[22] Gerber H. U., “Der Einfluss von Zins auf die Ruinwahrscheinlichkeit”, Mitteil. Verein. schweiz. Versich., 71:1 (1971), 63–70 | Zbl

[23] Gerber H. U., When does the surplus reach a given target?, Insurance Math. Econ., 9 (1990), 115–119 | DOI | MR | Zbl

[24] Gerber H. U., Shiu E. S. W., “The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin”, Insurance Math. Econ., 21 (1997), 129–137 | DOI | MR | Zbl

[25] Guérin H., Renaud J.-F., On distribution of cumulative Parisian ruin, arXiv: 1509.06857v1 [math.PR]

[26] Landriault D., Renaud J.-F., Zhou X., “Insurance risk models with Parisian implementation delays”, Methodol. Comput. Appl. Probab., 16:3 (2014), 583–607 | DOI | MR | Zbl

[27] Liu D., Liu Z., “Dividend problems with a barrier strategy in the dual risk model until bankruptcy”, J. Appl. Math., 2014, 184098

[28] Lkabous M. A., Czarna I., Renaud J.-F., Parisian ruin for a refracted Lévy process, arXiv: 1603.09324 [math.PR] | MR

[29] Quang P. D., “Ruin probability in a generalized risk process under interest force with homogenous Markov chain premiums”, Int. J. Stat. Probab., 2:4 (2013), 85–92 | DOI | MR

[30] Rachev S. T., Klebanov L., Stoyanov S. V., Fabozzi F., The Methods of Distances in the Theory of Probability and Statistics, Springer, New York, 2013 | MR | Zbl

[31] Rachev S. T., Stoyanov S. V., Fabozzi F. J., Advanced Stochastic Models, Risk Assessment, Portfolio Optimization, Wiley, Hoboken, 2008

[32] Saltelli A., Ratto M., Campolongo T., Cariboni J., Gatelli D., Saisana M., Tarantola S., Global Sensitivity Analysis. The Primer, Wiley, 2008 | MR | Zbl

[33] Sandström A., Handbook of Solvency for Actuaries and Risk Managers: Theory and Practice., Chapman and Hall/CRC Press, 2011 | Zbl