Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T04:42:03.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp Bounds for Sums of Dependent Risks

Part of: Mathematical economics Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Giovanni Puccetti  and
Ludger Rüschendorf
Giovanni Puccetti*
Affiliation:
University of Firenze
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗ Postal address: Department of Mathematics for Decision Theory, University of Firenze, via delle Pandette, 50127 Firenze, Italy. Email address: giovanni.puccetti@unifi.it
∗∗ Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany. Email address: ruschen@stochastik.uni-freiburg.de
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.

Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d=2 and, in some examples, for d≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case, F1=···=Fn, with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.

Keywords

Bounds for dependent risksFréchet boundmass transportation theory

MSC classification

Primary: 60E05: Distributions 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 42 - 53
Copyright
Copyright © Applied Probability Trust 2013 

References

Embrechts, P. and Puccetti, G. (2006a). Aggregating risk capital, with an application to operational risk. Geneva Risk Insurance Rev. 31, 71–90.Google Scholar
Embrechts, P. and Puccetti, G. (2006b). Bounds for functions of dependent risks. Finance Stoch. 10, 341–352.Google Scholar
Gaffke, N. and Rüschendorf, L. (1981). On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optimization 12, 123–135.Google Scholar
Makarov, G. D. (1981). Estimates for the distribution function of the sum of two random variables with given marginal distributions. Theory Prob. Appl. 26, 803–806.Google Scholar
Puccetti, G. and Rüschendorf, L. (2012a). Bounds for Joint portfolios of dependent risks. Statist. Risk Modeling 29, 107–132.Google Scholar
Puccetti, G. and Rüschendorf, L. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236, 1833–1840.Google Scholar
Puccetti, G., Wang, B. and Wang, R. (2012). Advances in complete mixability. J. Appl. Prob. 49, 430–440.Google Scholar
Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Z. Wahrscheinlichkeitsth. 57, 293–302.Google Scholar
Rüschendorf, L. (1982). Random variables with maximum sums. Adv. Appl. Prob. 14, 623–632.Google Scholar
Wang, B. and Wang, R. (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102, 1344–1360.Google Scholar
Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. To appear in Finance Stoch.Google Scholar