Bounds for sums of random variables when the marginal distributions and the variance of the sum are given

References

• Campana , A. 2007 . Tail value-at-risk for sums of non-independent random variables with a generalized Pareto distribution . GENEVA Risk and Insurance Review , 32 : 169 – 180 .
   Web of Science ®Google Scholar
• Cheung , K. C. 2007 . Characterizations of conditional comonotonicity . Journal of Applied Probability , 44 : 607 – 617 .
   Web of Science ®Google Scholar
• Cheung , K. C. 2008a . Improved convex upper bound via conditional comonotonicity . Insurance: Mathematics and Economics , 42 : 651 – 655 .
   Web of Science ®Google Scholar
• Cheung , K. C. 2008b . Characterization of comonotonicity using convex order . Insurance: Mathematics and Economics , 43 : 403 – 406 .
   Web of Science ®Google Scholar
• Cheung , K. C. 2009 . Upper comonotonicity . Insurance: Mathematics and Economics , 45 : 35 – 40 .
   Web of Science ®Google Scholar
• Cuesta , J. A. and Matran , C. 1989 . Notes on the Wasserstein metric in Hilbert spaces . Annals of Probability , 17 : 1264 – 1276 .
   Web of Science ®Google Scholar
• Deelstra , G. , Diallo , I. and Vanmaele , M. 2008 . Bounds for Asian basket options . Journal of Computational and Applied Mathematics , 218 : 215 – 228 .
   Web of Science ®Google Scholar
• Denneberg , D. 1994 . Non-additive measure and integral , Boston , MA : Kluwer Academic Publishers .
   Google Scholar
• Denuit , M. , Dhaene , J. , Goovaerts , M.J. & Kaas , R. ( 2005 ). Actuarial theory for dependent risks: measures, orders and models . Chichester , , UK : John Wiley .
   Google Scholar
• Dhaene , J. , Denuit , M. , Goovaerts , M.J. , Kaas , R. and Vyncke , D. 2002 . The concept of comonotonicity in actuarial science: theory . Insurance: Mathematics and Economics , 31 : 3 – 33 .
   Web of Science ®Google Scholar
• Dhaene , J. and Goovaerts , M. 1996 . Dependence of risks and stop-loss order . ASTIN Bulletin , 26 : 201 – 212 .
   Google Scholar
• Dhaene , J. , Goovaerts , M. & Vanduffel , S. ( 2008 ). Comonotonicity . In Encyclopedia of quantitative risk assessment and analysis , E. Melnick and B. Everitt , pp. 274 – 279 . Chichester , , UK : John Wiley .
   Google Scholar
• Dhaene , J. , Vanduffel , S. , Tang , Q. , Goovaerts , M. J. , Kaas , R. and Vyncke , D. 2006 . Risk measures and comonotonicity: a review . Stochastic Models , 22 : 573 – 606 .
   Web of Science ®Google Scholar
• Dong , J. , Cheung , K. C. & Yang , H. ( 2010 ). Upper comonotonicity and convex upper bounds for sums of random variables . Insurance: Mathematics and Economics 47 ( 2 ), 159 – 166 .
   Google Scholar
• Embrechts , P. and Pucetti , G. 2006 . Aggregating risk capital, with an application to operational risk . GENEVA Risk and Insurance Review , 31 : 71 – 90 .
   Web of Science ®Google Scholar
• Fang , K. T. , Kotz , S. and Ng , K. W. 1990 . Symmetric multivariate and related distributions , London : Chapman & Hall .
   Google Scholar
• Föllmer , H. and Schied , A. 2004 . Stochastic finance. An introduction in discrete time (2nd ed.) , Berlin : Walter de Gruyter .
   Google Scholar
• Jouini , E. and Napp , C. 2004 . Conditional comonotonicity . Decisions In Economics and Finance , 27 : 153 – 166 .
   Google Scholar
• Kaas , R. , Dhaene , J. , Vyncke , D. , Goovaerts , M. J. and Denuit , M. 2002 . A simple geometric proof that comonotonic risks have the convex-largest sum . ASTIN Bulletin , 32 : 71 – 80 .
   Google Scholar
• Müller , A. & Stoyan , D. ( 2002 ). Comparison methods for stochastic models and risks . Chichester , , UK : John Wiley .
   Google Scholar
• Valdez , E. , Dhaene , J. , Maj , M. and Vanduffel , S. 2009 . Bounds and approximations for sums of dependent log-elliptical random variables . Insurance: Mathematics and Economics , 44 : 385 – 397 .
   Web of Science ®Google Scholar
• Vandendorpe , A. , Ho , N. , Vanduffel , S. and Van Dooren , P. 2008 . On the parameterisation of the CreditRisk+ model for estimating credit portfolio risk . Insurance: Mathematics and Economics , 42 : 736 – 745 .
   Web of Science ®Google Scholar
• Vanduffel , S. , Chen , X. , Dhaene , J. , Goovaerts , M. and Henrard , L. 2008b . Optimal approximations for risk measures of sums of lognormals based on conditional expectations . Journal of Computational and Applied Mathematics , 221 ( 1 ) : 202 – 218 .
   Web of Science ®Google Scholar
• Vanduffel , S. , Dhaene , J. and Goovaerts , M. 2005 . On the evaluation of saving consumption plans . Journal of Pension Economics and Finance , 4 ( 1 ) : 17 – 30 .
   Google Scholar
• Vanduffel , S. , Shang , Z. , Shang , Z. , Henrard , L. , Dhaene , J. and Valdez , E. 2008a . Analytic bounds and approximations for Annuities and Asian options . Insurance: Mathematics and Economics , 42 ( 3 ) : 1109 – 1117 .
   Web of Science ®Google Scholar
• Vanmaele , M. , Deelstra , G. and Liinev , J. 2004 . Approximation of stop-loss premiums involving sums of lognormals by conditioning on two variables . Insurance: Mathematics and Economics , 35 : 343 – 368 .
   Web of Science ®Google Scholar