Asymptotic results for renewal risk models with risky investments. (English) Zbl 1250.91055
Summary: We consider a renewal jump-diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 60K05 | Renewal theory |
| 60J75 | Jump processes (MSC2010) |
Keywords:
renewal jump; diffusion process; ruin probability; Sparre Andersen risk model; investment; rational Laplace transform; regular variationReferences:
| [1] | Abate, J.; Whitt, W., Asymptotics for \(M / G / 1\) low-priority waiting-time tail probabilities, Queueing Sys., 25, 1-4 (1997) · Zbl 0894.60088 |
| [2] | Albrecher, H.; Constantinescu, C.; Pirsic, G.; Regensburger, G.; Rosenkranz, M., An algebraic operator approach to the analysis of Gerber-Shiu functions, Insurance Math. Econom., 46, 1, 42-51 (2010) · Zbl 1231.91135 |
| [3] | Asmussen, S.; Albrecher, H., (Ruin Probabilities. Ruin Probabilities, Advanced Series on Statistical Science & Applied Probability (2010), World Scientific Publishing Co. Inc.: World Scientific Publishing Co. Inc. River Edge, NJ) · Zbl 1247.91080 |
| [4] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., (Regular Variation. Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27 (1987), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0617.26001 |
| [5] | Collamore, J. F., Random recurrence equations and ruin in a Markov-dependent stochastic economic environment, Ann. Appl. Probab., 19, 4, 1404-1458 (2009) · Zbl 1176.60018 |
| [6] | C. Constantinescu, E. Thomann, Martingales for renewal jump-diffusion processes, Preprint, 2011.; C. Constantinescu, E. Thomann, Martingales for renewal jump-diffusion processes, Preprint, 2011. |
| [7] | Fedoryuk, M. V., Asymptotic Analysis (1993), Springer-Verlag: Springer-Verlag Berlin, Linear ordinary differential equations, Translated from the Russian by Andrew Rodick · Zbl 0782.34001 |
| [8] | Feller, W., An Introduction to Probability Theory and its Applications. vol. II (1971), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0219.60003 |
| [9] | Frolova, A.; Kabanov, Y.; Pergamenshchikov, S., In the insurance business risky investments are dangerous, Finance Stoch., 6, 2, 227-235 (2002) · Zbl 1002.91037 |
| [10] | Gaier, J.; Grandits, P., Ruin probabilities in the presence of regularly varying tails and optimal investment, Insurance Math. Econom., 30, 2, 211-217 (2002) · Zbl 1055.91049 |
| [11] | Gaier, J.; Grandits, P., Ruin probabilities and investment under interest force in the presence of regularly varying tails, Scand. Actuar. J., 4, 256-278 (2004) · Zbl 1091.62102 |
| [12] | Gaier, J.; Grandits, P.; Schachermayer, W., Asymptotic ruin probabilities and optimal investment, Ann. Appl. Probab., 13, 3, 1054-1076 (2003) · Zbl 1046.62113 |
| [13] | Gjessing, H. K.; Paulsen, J., Present value distributions with applications to ruin theory and stochastic equations, Stochastic Process. Appl., 71, 1, 123-144 (1997) · Zbl 0943.60098 |
| [14] | Goldie, C. M., Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., 1, 1, 126-166 (1991) · Zbl 0724.60076 |
| [15] | Grandits, P., A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market, Insurance Math. Econom., 34, 2, 297-305 (2004) · Zbl 1108.60061 |
| [16] | Hipp, C.; Plum, M., Optimal investment for insurers, Insurance Math. Econom., 27, 2, 215-228 (2000) · Zbl 1007.91025 |
| [17] | Kalashnikov, V.; Norberg, R., Power tailed ruin probabilities in the presence of risky investments, Stochastic Process. Appl., 98, 2, 211-228 (2002) · Zbl 1058.60095 |
| [18] | Klüppelberg, C.; Stadtmüller, U., Ruin probabilities in the presence of heavy-tails and interest rates, Scand. Actuar. J., 1, 49-58 (1998) · Zbl 1022.60083 |
| [19] | Kostadinova, R., Optimal investment for insurers when the stock price follows an exponential Lévy process, Insurance Math. Econom., 41, 2, 250-263 (2007) · Zbl 1193.91141 |
| [20] | Nyrhinen, H., Large deviations for the time of ruin, J. Appl. Probab., 36, 3, 733-746 (1999) · Zbl 0947.60048 |
| [21] | Nyrhinen, H., Finite and infinite time ruin probabilities in a stochastic economic environment, Stochastic Process. Appl., 92, 2, 265-285 (2001) · Zbl 1047.60040 |
| [22] | Paulsen, J., On Cramér-like asymptotics for risk processes with stochastic return on investments, Ann. Appl. Probab., 12, 4, 1247-1260 (2002) · Zbl 1019.60041 |
| [23] | Paulsen, J., Ruin models with investment income, Probab. Surv., 5, 416-434 (2008) · Zbl 1189.91077 |
| [24] | Paulsen, J.; Gjessing, H. K., Ruin theory with stochastic return on investments, Adv. in Appl. Probab., 29, 4, 965-985 (1997) · Zbl 0892.90046 |
| [25] | Pergamenshchikov, S., Erratum to: “Ruin probability in the presence of risk investments” [Stochastic Process Appl. 116 (2006) 267-278], Stochastic Process. Appl., 119, 1, 305-306 (2009) · Zbl 1165.60334 |
| [26] | Pergamenshchikov, S.; Zeitouny, O., Ruin probability in the presence of risky investments, Stochastic Process. Appl., 116, 2, 267-278 (2006) · Zbl 1088.60076 |
| [27] | Tang, Q.; Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl., 108, 2, 299-325 (2003) · Zbl 1075.91563 |
| [28] | Tang, Q.; Tsitsiashvili, G., Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments, Adv. in Appl. Probab., 36, 4, 1278-1299 (2004) · Zbl 1095.91040 |
| [29] | Tang, Q.; Wei, L., Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence, Insurance Math. Econom., 46, 1, 19-31 (2010) · Zbl 1231.91243 |
| [30] | Teugels, J. L.; Willmot, G., Approximations for stop-loss premiums, Insurance Math. Econom., 6, 3, 195-202 (1987) · Zbl 0624.62097 |
| [31] | Thomann, E.; Waymire, E., Contingent claims on assets with conversion costs, J. Statist. Plann. Inference, 113, 403-417 (2003) · Zbl 1071.91023 |
| [32] | Wang, G.; Wu, R., Distributions for the risk process with a stochastic return on investments, Stochastic Process. Appl., 95, 2, 329-341 (2001) · Zbl 1064.91051 |
| [33] | Wei, L., Ruin probability of the renewal model with risky investment and large claims, Sci. China Ser. A, 52, 7 (2009) · Zbl 1187.60081 |
| [34] | Yuen, K. C.; Wang, G.; Wu, R., On the renewal risk process with stochastic interest, Stochastic Process. Appl., 116, 10, 1496-1510 (2006) · Zbl 1109.60071 |
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