In this paper we propose a quadratic predictor of the compound discounted renewal aggregate claims when taking into account dependence within the inter-occurrence times. We compare the accuracy of the proposed quadratic predictor to the simulated value of that sum, by using specific mixture of exponential distributions to define the dependence structure between the inter-occurrence times.
AdékambiF., Annual Proceedings of the South African Statistical Association Conference, Volume 2017 Number Congress 1, Dec 2017, pp. 1–7.
2.
AdékambiF. and DziwaS., Annual Proceedings of the South African Statistical Association Conference, Congress 1, Nov 2016, pp. 1–8.
3.
AlbrecherH.ContantinescuC. and LoiselS., Explicit ruin formulas for models with dependence among risks, Insurance: Mathematics and Economics48(2) (2011), 265–270.
4.
AlbrecherH. and TeugelsJ.L., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability43(1) (2006), 257–273.
5.
AlbrecherHansjörg and KortschakDominik, On ruin probability and aggregate claim representations for Pareto claim size distributions, Insurance: Mathematics and Economics, Elsevier45(3) (2009), 362–373.
6.
BargèsM.CossetteH.LoiselS. and MarceauE., On the moments of the aggregate discounted claims with dependence introduced by a FGM copula, ASTIN Bulletin, Cambridge University Press (CUP)41(1) (2011), 215–238 , hal-00426502.
7.
BoogaertP.HaezendonckJ. and DelbaenF., Limit theorems for the present value of the surplus of an insurance portfolio, Insurance: Mathematics and Economics7(2) (1988), 131–138.
8.
CaiJ. and DicksonD.C.M., Upper bounds for ultimate ruin probability in the Sparre Andersen model with interest, Insurance: Mathematics and Economics32(1) (2003), 61–71.
9.
CairnsA.J.G., Interest Rates Models, Princeton University Press, Princeton, NJ, 2004.
10.
CressieN.DavisA.S.FolksJ.L. and FolksJ.L., The moment-generating function and negative integer moments, The American Statistician35 (1981), 148–150.
11.
CossetteHélèneMarceauEtienneMtalaiItre and VeilleuxDéry, Dependent Risk Models with Archimedean Copulas: A Computational Strategy Based on Common Mixtures and Applications (October 30, 2017). Available at SSRN: https://ssrn.com/abstract=2957181 or doi:10.2139/ssrn.2957181.
12.
DelbaenF. and HaezendonckJ., Classical risk theory in an economic environment, Insurance: Mathematics and Economics6 (1987), 85–116.
13.
FellerW., An Introduction to Probability Theory and its Applications, Second ed. Vol. 1, John Wiley, New York, 1971.
14.
GerberH.U., Deir Einfluss von Zins auf Ruinwarhrscheinlichkeit, Mitteilungen Vereinigung Schweizerische Versicherungsmathematiker71 (1971), 63–70.
15.
GleserL.-J., The gamma distribution as a mixture of exponential distributions, The American Statistician43 (1989), 115–117.
16.
JangJ.W., Martingale approach for moments of discounted aggregate claims, The Journal of Risk and Insurance71(2) (2004), 201–211.
17.
KaratzasI. and ShreveS.E., Brownian Motion and Stochastic Calculus, 2nd ed. Springer-Verlag, New York, 1991.
18.
KimB. and KimH.-S., Moments of claims in a Markovian environment, Insurance: Mathematics and Economics40(3) (2007), 485–497.
19.
LéveilléG. and AdékambiF., Covariance of discounted compound renewal sums with a stochastic interest rate, Scandinavian Actuarial Journal2 (2011), 138–153.
20.
LéveilléG. and AdékambiF., Joint moments of discounted compound renewal sums, Scandinavian Actuarial Journal1 (2012), 40–55.
21.
LéveilléG. and GarridoJ., Moments of compound renewal sums with discounted claims, Insurance: Mathematics and Economics28 (2001), 217–231.
22.
LéveilléG. and GarridoJ., Recursive moments of compound renewal sums with discounted claims, Scandinavian Actuarial Journal2 (2001), 98–110.
23.
LéveilléG. and GarridoJ., Inflation impact on aggregate claims, Encyclopedia of Actuarial Science (2004).
24.
LéveilléG.GarridoJ. and WangY.F., Moment generating function of compound renewal sums with discounted claims, Scandinavian Actuarial Journal1 (2009), 1–20.
25.
OksendalB., Stochastic Differential Equation, 3rd ed. Springer-Verlag, New York, 1992.
26.
SundtB. and TeugelsJ., Ruin estimate under interest force, Insurance: Mathematics and Economics16 (1995), 7–22.
27.
SarabiaJ.M.Gómez-DénizE.PrietoF. and JordáV., Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions - arXiv preprint arXiv:1705.00289, 2017.
28.
SendovaK.P. and ZitikisR., Aggregate claims when their sizes and arrival times are dependent and governed by a general point process - arXiv preprint arXiv:1104.4742, 2011.
29.
SklarA., Fonctions de répartition à n dimensions et leurs marges, Publications de l'Institut de Statistique de l'Université de Paris8 (1959), 229–231.
30.
TaylorG.C., Probability of ruin under inflationary conditions or under experience rating, Astin Bulletin10 (1979), 149–162.
31.
WatersH., Probability of ruin with claims cost inflation, Scandinavian Actuarial Journal2 (1989), 148–164.
32.
WillmotG.E., The total claims distribution under inflationary conditions, Scandinavian Actuarial Journal10 (1989), 1–12.
33.
WhitmoreG.A., Inverse Gaussian mixtures of exponential distributions, unpublished paper, McGill University, Faculty of management, 1988.
34.
WhitmoreG.A. and LeeM.-L.T., A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials, Technometrics33 (1991), 39–50.
