Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof
Abstract
:1. Introduction
- (i)
- mutually exclusive from below if for all ;
- (ii)
- mutually exclusive from above if for all .
2. Proof of Theorem 1
3. Proof of Theorem 2
- (1)
- .
- (2)
- for all convex functions v, such that the expectations exist.
- (3)
- for all functions v with , such that the expectations exist.
Acknowledgments
Author Contributions
Conflicts of Interest
References
- M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial Theory for Dependent Risks: Measures, Orders and Models. Chichester, UK: John Wiley & Sons, 2005. [Google Scholar]
- M. Shaked, and J.G. Shanthikumar. Stochastic Orders. New York, NY, USA: Springer, 2007. [Google Scholar]
- M.E. Yaari. “The dual theory of choice under risk.” Econometrica 55 (1987): 95–115. [Google Scholar] [CrossRef]
- D. Schmeidler. “Integral representation without additivity.” Proc. Am. Math. Soc. 97 (1986): 255–261. [Google Scholar] [CrossRef]
- J. Dhaene, M. Denuit, M.J. Goovaerts, R. Kaas, and D. Vyncke. “The concept of comonotonicity in actuarial science and finance: Theory.” Insur. Math. Econ. 31 (2002): 3–33. [Google Scholar] [CrossRef]
- J. Dhaene, M. Denuit, M.J. Goovaerts, R. Kaas, and D. Vyncke. “The concept of comonotonicity in actuarial science and finance: Applications.” Insur. Math. Econ. 31 (2002): 133–161. [Google Scholar] [CrossRef]
- G. Deelstra, J. Dhaene, and M. Vanmaele. “An overview of comonotonicity and its applications in finance and insurance.” In Advanced Mathematical Methods for Finance. Edited by B. Oksendal and G. Nunno. Heidelberg, Germany: Springer, 2010, pp. 155–179. [Google Scholar]
- I. Meilijson, and A. Nadas. “Convex majorization with an application to the length of critical paths.” J. Appl. Probab. 16 (1979): 671–677. [Google Scholar] [CrossRef]
- A.H. Tchen. “Inequalities for distributions with given marginals.” Ann. Probab. 8 (1980): 814–827. [Google Scholar] [CrossRef]
- L. Rus̎chendorf. “Solution of a statistical optimization problem by rearrangement methods.” Metrika 30 (1983): 55–61. [Google Scholar] [CrossRef]
- J. Dhaene, and M.J. Goovaerts. “Dependency of risks and stop-loss order.” ASTIN Bull. 26 (1996): 201–212. [Google Scholar] [CrossRef]
- J. Dhaene, and M.J. Goovaerts. “On the dependency of risks in the individual life model.” Insur. Math. Econ. 19 (1997): 243–253. [Google Scholar] [CrossRef]
- A. Müller. “Stop-loss order for portfolios of dependent risks.” Insur. Math. Econ. 21 (1997): 219–223. [Google Scholar] [CrossRef]
- R. Kaas, J. Dhaene, D. Vyncke, M. Goovaerts, and M. Denuit. “A simple geometric proof that comonotonic risks have the convex-largest sum.” ASTIN Bull. 32 (2002): 71–80. [Google Scholar] [CrossRef]
- K.C. Cheung. “Comonotonic convex upper bound and majorization.” Insur. Math. Econ. 47 (2010): 154–158. [Google Scholar] [CrossRef]
- K.C. Cheung. “Characterization of comonotonicity using convex order.” Insur. Math. Econ. 43 (2008): 403–406. [Google Scholar] [CrossRef]
- K.C. Cheung. “Characterizing a comonotonic random vector by the distribution of the sum of its components.” Insur. Math. Econ. 47 (2010): 130–136. [Google Scholar] [CrossRef]
- T.T. Mao, and T.Z. Hu. “A new proof of Cheung’s characterization of comonotonicity.” Insur. Math. Econ. 48 (2011): 214–216. [Google Scholar] [CrossRef]
- G. Puccetti, and R. Ruodu Wang. “Extremal dependence concepts.” Stat. Sci. 30 (2015): 485–517. [Google Scholar] [CrossRef]
- K.C. Cheung, and A. Lo. “Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order.” Insur. Math. Econ. 53 (2013): 334–342. [Google Scholar] [CrossRef]
- C. Bernard, X. Jiang, and R. Wang. “Risk aggregation with dependence uncertainty.” Insur. Math. Econ. 54 (2014): 93–108. [Google Scholar] [CrossRef]
- B. Wang, and R. Wang. “The complete mixability and convex minimization problems for monotone marginal distributions.” J. Multivar. Anal. 102 (2011): 1344–1360. [Google Scholar] [CrossRef]
- B. Wang, and R. Wang. “Joint mixability.” Math. Op. Res. 41 (2016): 808–826. [Google Scholar] [CrossRef]
- E. Jakobsons, X. Han, and R. Wang. “General convex order on risk aggregation.” Scand. Actuar. J. 8 (2016): 713–740. [Google Scholar] [CrossRef]
- J. Dhaene, and M. Denuit. “The safest dependence structure among risks.” Insur. Math. Econ. 25 (1999): 11–21. [Google Scholar] [CrossRef]
- T.Z. Hu, and Z.Q. Wang. “On the dependence of risks and the stop-loss premiums.” Insur. Math. Econ. 24 (1999): 323–332. [Google Scholar] [CrossRef]
- K.C. Cheung, and A. Lo. “Characterizing mutual exclusivity as the strongest negative multivariate dependence structure.” Insur. Math. Econ. 55 (2014): 180–190. [Google Scholar] [CrossRef] [Green Version]
- J. Dhaene, A. Kukush, D. Linders, and Q. Tang. “Remarks on quantiles and distortion risk measures.” Eur. Actuar. J. 2 (2012): 319–328. [Google Scholar] [CrossRef]
- J. Dhaene, S. Wang, V. Young, and M. Goovaerts. “Comonotonicity and maximal stop-loss premiums.” Bull. Swiss Assoc. Actuar. 2 (2000): 99–113. [Google Scholar]
- J. Dhaene, S. Vanduffel, Q. Tang, M. Goovaerts, R. Kaas, and D. Vyncke. “Risk measures and comonotonicity: A review.” Stoch. Models 22 (2006): 573–606. [Google Scholar] [CrossRef]
- D. Denneberg. Non-Additive Measure and Integral. Dordrecht, The Netherlands: Kluwer Academic Publilshers, 1994. [Google Scholar]
- S. Wang, and J. Dhaene. “Comonotonicity, correlation order and premium principles.” Insur. Math. Econ. 22 (1998): 235–242. [Google Scholar] [CrossRef]
- K.C. Cheung, and A. Lo. “General lower bounds on convex functionals of aggregate sums.” Insur. Math. Econ. 53 (2013): 884–896. [Google Scholar] [CrossRef]
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yin, C.; Zhu, D. Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof. Risks 2016, 4, 34. https://doi.org/10.3390/risks4040034
Yin C, Zhu D. Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof. Risks. 2016; 4(4):34. https://doi.org/10.3390/risks4040034
Chicago/Turabian StyleYin, Chuancun, and Dan Zhu. 2016. "Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof" Risks 4, no. 4: 34. https://doi.org/10.3390/risks4040034