Skip to main content
SpringerLink
Log in

Second Order Asymptotics of Aggregated Log-Elliptical Risk

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions satisfied in particular by log-normal risks. Given the wide range of applications of the log-normal model in finance and insurance our result is of interest for both rare-event simulations and numerical calculations. We present numerical examples which illustrate that the second order approximation derived in this paper significantly improves over the first order approximation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albrecher H, Asmussen S, Kortschak D (2006) Tail asymptotics for the sum of two heavy tailed dependent risks. Extremes 9:107–130

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S, Blanchet J, Juneja S, Rojas-Nandayapa L (2011) Efficient simulation of tail probabilities of sums of correlated lognormals. Ann Oper Res 189:5–23

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S, Rojas-Nandayapa L (2008) Sums of dependent log-Gaussian random variables with Gaussian copula. Statist Probab Lett 78:2709–2714

    Article  MathSciNet  MATH  Google Scholar 

  • Berman MS (1992) Sojourns and extremes of stochastic processes. Wadsworth & Brooks/ Cole, Boston

    MATH  Google Scholar 

  • Bingham NN, Goldie CM, Teugels J (1987) Regular variation. Cambridge University Press, Cambridge, UK

    Book  MATH  Google Scholar 

  • Cambanis S, Huang S, Simons G (1981) On the theory of elliptically contoured distributions. J Multivariate Anal 11:368–385

    Article  MathSciNet  MATH  Google Scholar 

  • Foss S, Richards A (2010) On sums of conditionally independent subexponential random variables. Math Oper Res 35:102–119

    Article  MathSciNet  MATH  Google Scholar 

  • Embrechts P, Klüppelberg C, Mikosch T (1997) Modeling extremal events for inance and insurance. Springer, Berlin

    Book  Google Scholar 

  • Hashorva E (2013) Exact tail asymptotics of aggregated parametrised risk. J Math Anal Appl 400:187–199

    Article  MathSciNet  MATH  Google Scholar 

  • Hashorva E (2012) Exact tail asymptotics in bivariate scale mixture models. Extremes 15:109–128

    Article  MathSciNet  Google Scholar 

  • Kortschak D (2011) Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks. Extremes 15:353–388

    Article  MathSciNet  Google Scholar 

  • Kortschak D, Hashorva E (2013) Efficient simulation of tail probabilities for sums of log-elliptical risks. J Comput Appl Math 247:53–67

    Article  MathSciNet  MATH  Google Scholar 

  • Mitra A, Resncik SI (2009) Aggregation of rapidly varying risks and asymptotic independence. Adv Appl Probab 41:797–828

    Article  MATH  Google Scholar 

  • Rojas-Nandayapa L (2008) Risk probabilities: asymptotics and simulation. PhD Thesis, Aarhus University, Denmark

  • Resnick SI (1987) Extreme values, regular variation and point processes. Springer, New York

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Lyon, 69622, Lyon, France

    Dominik Kortschak

  2. Laboratoire SAF, EA 2429, Institut de Science Financière et d’Assurances, Université Lyon 1, 50 Avenue Tony Garnier, 69007, Lyon, France

    Dominik Kortschak

  3. Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, 1015, Lausanne, Switzerland

    Enkelejd Hashorva

Authors
  1. Dominik Kortschak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enkelejd Hashorva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik Kortschak.

Additional information

D. Kortschak was supported by the the MIRACCLE-GICC project and the Chaire d’excellence “Generali—Actuariat responsable: gestion des risques naturels et changements climatiques.” E. Hashorva kindly acknowledges partial support by Swiss National Science Foundation Grants 200021-134785 and 200021-1401633/1 and by RARE -318984, a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kortschak, D., Hashorva, E. Second Order Asymptotics of Aggregated Log-Elliptical Risk. Methodol Comput Appl Probab 16, 969–985 (2014). https://doi.org/10.1007/s11009-013-9356-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9356-5

Keywords

AMS 2000 Subject Classification

Search

Navigation