Investigation of a Non-Linear Cramér-Lundberg Risk Model

Yıl 2022, Cilt: 6 Sayı: 1, 1065 - 1075, 30.06.2022

Zulfiye Hanalioglu Yusup Allyyev Tahir Khanıyev

Öz

In this study, a non-linear version of a Cramér-Lundberg risk model is examined. The objective of this work is to evaluate the ruin probability of a non-linear risk model. The classical linear Cramér-Lundberg model has been widely studied in the literature. However, the linear model is not always realistic. Because an insurance company's premium income cannot always increase linearly. Therefore, it is recommended to adapt premium income as a function which increases monotonically and yet its rate of growth decreases over time. Thus, to account for this, a more realistic non-linear mathematical model has been constructed and investigated, when the premium income function is p(t)=c√t. Then Lundberg type upper bound was calculated for the ruin probability for the model under investigation.

Anahtar Kelimeler

Cramér-Lundberg risk model, Ruin probability, Non-linear risk model, Lundberg type upper bound, Generalized Lundberg coefficient

Kaynakça

• [1] Asmussen, S., Rolski, T., (1994), “Risk Theory in a Periodic Environment: The Cramér-Lundberg Approximation and Lundberg's Inequality”, Mathematics of Operations Research, 19 (2), 410-433.
• [2] Boikov, A.V., (2002), “The Cramér-Lundberg model with stochastic premium process”, Theory of Probability and Applications, 47, 489-493.
• [3] Chadjiconstantinidis, S., Politis, K., (2007), "Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model", Insurance: Mathematics and Economics, 41(1), 41-52.
• [4] Cohen, A., R.Young, V., (2020), “Rate of convergence of the probability of ruin in the Cramér-Lundberg model to its diffusion approximation”, Insurance: Mathematics and Economics, 93, 333-340.
• [5] Constantinescu, C., Samorodnitsky, G., Zhu, W., (2018), “Ruin probabilities in classical risk models with gamma claims”, Scandinavian Actuarial Journal, 2018(7), 555-575.
• [6] Cramér, H., (1930), “On the mathematical theory of risk”, Skandinavia Jubilee Volume, Stockholm. Reprinted in: martin-Löf, A. (Ed.) Cramér, H. (1994) Collected Works. Springer, 155-166.
• [7] Gaier, J., Grandits, P., Schachermayer, W., (2003), “Asymptotic Ruin Probabilities and Optimal Investment”, The Annals of Applied Probability, 13 (3), 1054-1076.
• [8] Gauchonab, R., Loisela, S., Rullièrea, J., Trufinc, J., (2020), “Optimal prevention strategies in the classical risk model”, Insurance: Mathematics and Economics, 91, 202-208.
• [9] Gerber, H.U. (1988), “Mathematical fun with ruin theory”, Insurance: Mathematics and Economics, 7(1), 15-23.
• [10] Kaas R., Goovaerts M., Dhaene J., Denuit M., (2001), “Modern Actuarial Risk Theory”, Kluwer, Boston.
• [11] Lundberg, F., (1903), “Approximerad framställning av sannolikhetsfunktionen”, Återförsäkring av kollektivrisker. Akad. Afhandling. Almqvist och Wiksell, Uppsala, 7-9.
• [12] Malinovskii, V.K, (2014), “Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk”, Insurance: Mathematics and Economics, 55, 301-309.
• [13] Mikosch, T. (2004), “Non-life insurance mathematics: An Introduction with Stochastic Processes”, Springer-Verlag, Berlin.
• [14] Mishura, Y., Perestyuk, M., Ragulina, O., (2014), “Ruin probability in a risk model with variable premium intensity and risky investments”, Opuscula Mathematica, 35(2).
• [15] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., (1999), “Stochastic Processes for Insurance and Finance”, Wiley, New York.
• [16] Straub E. (1988), “Non-Life Insurance Methematics”, Springer, New York.
• [17] Temnov, G., (2014), “Risk Models with Stochastic Premium and Ruin Probability Estimation”, Journal of Mathematical Sciences, 196, 84-96.
• [18] Willmot, G.E., Lin, X.S., (2001), “Lundberg Approximations for Compound Distributions with Insurance Applications”, Springer, Berlin.
• [19] Yang H. (1998), “Non-exponential Bounds for Ruin Probability with Interest Effect Included”, Scandinavian Actuarial Journal, 1999(1), 66-79.
• [20] Zhang, Z., Yang, H., (2009), “On a risk model with stochastic premiums income and dependence between income and loss”, Journal of Computational and Applied Mathematics, 234(1), 44-57.