Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T13:38:01.484Z Has data issue: false hasContentIssue false
  • English
  • Français

The Gerber-shiu Expected Discounted Penalty-reward Function under an Affine Jump-diffusion Model

Published online by Cambridge University Press:  17 April 2015

Florin Avram  and
Miguel Usabel
Florin Avram
Affiliation:
Department of Mathematics Université de Pau, France, E-Mail: Florin.Avram@univ-pau.fr
Miguel Usabel
Affiliation:
Department of Business Administration, Universidad Carlos III de Madrid, Spain, E-Mail: usabel@emp.uc3m.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a unified analytical treatment of first passage problems under an affine state-dependent jump-diffusion model (with drift and volatility depending linearly on the state). Our proposed model, that generalizes several previously studied cases, may be used for example for obtaining probabilities of ruin in the presence of interest rates under the rational investement strategies proposed by Berk & Green (2004).

Keywords

First passage problemsrisk processstochastic rates of interestruin with interestaffine jump-diffusion modelspenalty/reward functions at ruin
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 38 , Issue 2 , November 2008 , pp. 461 - 481
Copyright
Copyright © ASTIN Bulletin 2008

References

Asmussen, S. and Bladt, M. (1996) Phase-type distributions and Risk Processes with state-dependent premiums. Scand. Act. Journal 1996, 19–36.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. (1970) Handbook of Mathematical Functions. 9th printing.Google Scholar
Berk, J. and Green, R. (2004) Mutual Funds flows and performance in rational markets. Journal of Political Economy 112(6), 1269–1295.CrossRefGoogle Scholar
Cai, J. and Dickson, D.C.M. (2002) On the expected discounted penalty function at ruin of a surplus process with interest. Insurance: Mathematics and Economics 30, 389–404.Google Scholar
Cai, J. (2004). Ruin probabilities and penalty functions with stochastic rates of interest. Stochastic Processes and Their Applications 112, 53–78.CrossRefGoogle Scholar
Cai, J. and Yang, H.L. (2005) Ruin in the perturbed compound Poisson risk process under interest force. Advances in Applied Probability 37, 819–835.CrossRefGoogle Scholar
Delbaen, F. and Haezendonck, J. (1987) Classical risk theory in an econmic environment. Insurance: Mathematics and Economics 6, 85–116.Google Scholar
Dickson, D.C.M. and Waters, H.R. (1999) Ruin probability with compounding assets. Insurance: Mathematics and Economics 25, 49–62.Google Scholar
Duffie, D., Filipovic, D. and Schachermayer, W. (2003) Affine processes and aplications in finance. Annals of Applied Probability 13(3), 984–1053.CrossRefGoogle Scholar
Embrechts, P. and Schmidli, H. (1994). Ruin estimations for a general insurance risk model. Advances in applied probability 26, 404–422.CrossRefGoogle Scholar
Gaier, J. and Grandits, P. (2004) Ruin probabilities and investment under interest force in the presence of regularly varying tails. Scand. Actuarial Journal 4, 256–278.CrossRefGoogle Scholar
Garrido, J. (1989) Stochastic diffrential equations for compound risk reserves. Insurance: Mathematics and Economics 8, 165–173.Google Scholar
Gerber, H. and Shiu, E. (1998) On the time value of ruin. North American Actuarial Journal 2, 48–78.CrossRefGoogle Scholar
Gerber, H. and Yang, H. (2007) Absolute ruin probabilities ina jump diffusion risk model with investments. North American Actuarial Journal 11(3), 159–169.CrossRefGoogle Scholar
Göing-Jeaschke, A. and Yor, M. (2003) A survey and some generalizations of Bessel processes. Bernoulli 9(2), 313–349.Google Scholar
Gradshteyn, I.S. and Ryzhik, I.M. (1994) Table of Integrals, Series and Products. Fifth edition. Academic Press.Google Scholar
Kalashnikov, V. and Norberg, R. (2002) Power tailed ruin probabilities in the presence of risky investments. Stochastic Processes and Their Applications 98, 211–228.CrossRefGoogle Scholar
Ma, J. and Sun, X. (2003) Ruin probabilities for insurance models involving investments. Scandinavian Actuarial Journal, 217–237.Google Scholar
Norberg, R. (1999) Ruin problems with assets and liabilities of diffusion type. Stochastic processes and their applications 81, 255–269.CrossRefGoogle Scholar
Novikov, A. (2003) Martingales and First-Passage Times for Ornstein-Uhlenbeck Processes with a Jump Component. Theory of Probability and Its Applications 48(2), 288–303.CrossRefGoogle Scholar
Paulsen, J. (1993) Risk theory in a stochastic environment. Stochastic processes and their applications 21, 327–361.CrossRefGoogle Scholar
Paulsen, J. and Gjessing, H.K. (1997a) Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics 20, 215–223.Google Scholar
Paulsen, J. and Gjessing, H.K. (1997b) Ruin theory with stochastic return on investments. Advances in Applied Probability 29, 965–985.Google Scholar
Paulsen, J. (1998) Risk theory with compounding assets – a survey. Insurance: Mathematics and Economics 22, 3–16.Google Scholar
Paulsen, J., Kasozi, J. and Steigen, A. (2005) A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments. Insurance: Mathematics and Economics 36, 399–420.Google Scholar
Peters, K. (1994) Exact and asymptotic solutions for the time-dependent problem in collective ruin I. SIAM J. Appl. Math. 54, 1761.Google Scholar
Segerdahl, C.O. (1942) Er einige risikotheoretische Fragestellungen. Skand. Aktuar Tidskr. 25, 43–83.Google Scholar
Sundt, B. and Teugels, J.L. (1995) Ruin estimates under interest force. Insurance: Mathematics and Economics 16, 7–22.Google Scholar
Wang, G. and Wu, R. (2001) Distributions for the risk process with stochastic return on investments. Stochastic processes and their applications 95, 329–341.CrossRefGoogle Scholar
Willmot, G. and Dickson, D. (2003) The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insurance: Mathematics and Economics 32, 403–411.Google Scholar
Yuen, K.C., Wang, G. and Ng, W.K. (2004) Ruin probabilities for a risk process with stochastic return on investments. Stochastic processes and their applications 110, 259–274.CrossRefGoogle Scholar
Yuen, K.C. and Wang, G. (2005) Some ruin problems for a risk process with stochastic interest. North American Actuarial Journal 9(3), 129–142.CrossRefGoogle Scholar