Journal of Applied Probability

Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments

Xuemiao Hao and Qihe Tang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y0=x > 0, evolves according to dYt= Yt- dRt -dPt for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.

Article information

J. Appl. Probab., Volume 49, Number 4 (2012), 939-953.

First available in Project Euclid: 5 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91B30: Risk theory, insurance
Secondary: 60G51: Processes with independent increments; Lévy processes 91B28

(extended) regular variation finite-time and infinite-time ruin probabilities Lévy process stochastic difference equation tail probability


Hao, Xuemiao; Tang, Qihe. Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments. J. Appl. Probab. 49 (2012), no. 4, 939--953. doi:10.1239/jap/1354716649.

Export citation


  • Albrecher, H., Constantinescu, C. and Thomann, E. (2012). Asymptotic results for renewal risk models with risky investments. Stoch. Process. Appl. 122, 3767–3789.
  • Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.
  • Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.
  • Chen, Y. and Yuen, K. C. (2009). Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25, 76–89.
  • Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.
  • Frolova, A., Kabanov, Y. and Pergamenshchikov, S. (2002). In the insurance business risky investments are dangerous. Finance Stoch. 6, 227–235.
  • Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.
  • Grey, D. R. (1994). Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Prob. 4, 169–183.
  • Heyde, C. C. and Wang, D. (2009). Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv. Appl. Prob. 41, 206–224.
  • Hult, H. and Lindskog, F. (2011). Ruin probabilities under general investments and heavy-tailed claims. Finance Stoch. 15, 243–265.
  • Kalashnikov, V. and Konstantinides, D. (2000). Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom. 27, 145–149.
  • Kalashnikov, V. and Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228.
  • Klüppelberg, C. and Kostadinova, R. (2008). Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42, 560–577.
  • Klüppelberg, C. and Stadtmüller, U. (1998). Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuarial J. 1998, 49–58.
  • Konstantinides, D., Tang, Q. and Tsitsiashvili, G. (2002). Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Math. Econom. 31, 447–460.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Liptser, R. S. and Shiryayev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht.
  • Olvera-Cravioto, M. (2012). Asymptotics for weighted random sums. Adv. Appl. Prob. 44, 1142–1172.
  • Paulsen, J. (1998a). Ruin theory with compounding assets–a survey. The interplay between insurance, finance and control. Insurance Math. Econom. 22, 3–16.
  • Paulsen, J. (1998b). Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Process. Appl. 75, 135–148.
  • Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Prob. 12, 1247–1260.
  • Paulsen, J. (2008). Ruin models with investment income. Prob. Surveys 5, 416–434.
  • Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
  • Resnick, S. I. and Willekens, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statist. Stoch. Models 7, 511–525.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.
  • Tang, Q. (2005). The finite-time ruin probability of the compound Poisson model with constant interest force. J. Appl. Prob. 42, 608–619.
  • Tang, Q. and Tsitsiashvili, G. (2003). 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.
  • Tang, Q., Wang, G. and Yuen, K. C. (2010). Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insurance Math. Econom. 46, 362–370.
  • Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.
  • Zhang, Y., Shen, X. and Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119, 655–675. \endharvreferences