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Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims

Part of: Stochastic processes Applications Mathematical economics

Published online by Cambridge University Press:  01 July 2016

C. C. Heyde  and
Dingcheng Wang
C. C. Heyde
Affiliation:
Australian National University and Columbia University
Dingcheng Wang*
Affiliation:
Australian National University and University of Electronic Science and Technology of China
*
∗∗ Postal address: Center of Financial Mathematics, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: dingcheng.wang@anu.edu.au
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Abstract

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By expressing the discounted net loss process as a randomly weighted sum, we investigate the finite-time ruin probabilities for the Poisson risk model with an exponential Lévy process investment return and heavy-tailed claims. It is found that in finite time, however, the extreme of insurance risk dominates the extreme of financial risk, but, for the case of dangerous investment (see Klüppelberg and Kostadinova (2008) for an accurate definition of dangerous investment), the extreme of financial risk has more and more of an effect on the total risk, and as time passes, the extreme of financial risk finally dominates the extreme of insurance risk.

Keywords

Finite-time ruin probabilityinvestment returnLévy processPoisson risk modelself-financing portfolioregularly varying tailclaim

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 206 - 224
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

∗

C. C. Heyde died in March 2008.

References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes, eds Barndorff-Nielsen, O. E. et al., Birkhäuser, Boston, MA, pp. 41–55.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.Google Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Embrechts, P., Kläppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.Google Scholar
Emmer, S. and Klüppelberg, C. (2004). Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8, 17–44.Google Scholar
Emmer, S., Klüppelberg, C. and Korn, R. (2001). Optimal portfolios with bounded capital at risk. Math. Finance 11, 365–384.Google Scholar
Frolova, A., Kabanov, Y. and Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Finance Stoch. 6, 227–235.Google Scholar
Gaier, J. and Grandits, P. (2002). Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance Math. Econom. 30, 211–217.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
Hurst, S. R., Platen, E. and Rachev, S. T. (1999). Option pricing for a logstable asset price model. Math. Comput. Modelling 29, 105–119.CrossRefGoogle Scholar
Kalashnikov, V. and Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228.Google Scholar
Klüppelberg, C. and Konstadinova, R. (2008). Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42, 560–577.Google Scholar
Konstantinides, D. G. and Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Prob. 33, 1992–1992.Google Scholar
Korn, K. (1997). Optimal Portfolios. World Scientific, Singapore.CrossRefGoogle Scholar
Kou, S. G. (2002). A Jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101.Google Scholar
Lindner, A. and Maller, R. (2005). Levy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722.Google Scholar
Madan, D. B. and Seneta, E. (1990). The variance gamma (VG) model for share market returns. J. Business 63, 511–524.Google Scholar
Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investements. Ann. Appl. Prob. 12, 1247–1260.Google Scholar
Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29, 965–985.CrossRefGoogle Scholar
Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar
Tang, Q. and Tsitsiashvili, G. (2003a). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299–325.Google Scholar
Tang, Q. and Tsitsiashvili, G. (2003b). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6, 171–188.Google Scholar
Tang, Q. and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob. 36, 1278–1299.Google Scholar
Wang, D. C. (2008). Finite-time ruin probability with heavy-tailed claims and constant interest rate. Stoch. Models 24, 41–57.CrossRefGoogle Scholar