×

Upper and lower bounds for sums of random variables. (English) Zbl 0989.60019

In actuarial sciences, the stop-loss order and convex order concepts are closely related, hence it seems naturally to study upper and lower bounds for sums \(X_1 + X_2 + \ldots + X_n\) of dependent random variables in the sense of convex order. The paper gives such an answer, under mild hypothesis: the marginal distribution of each \(X_i\) is known, and there exists a random variable \(Z\) such that the distribution functions of \(X_i\), given \(Z=z\), are known. A numerical illustration is given in the case of lognormal random variables.

MSC:

60E15 Inequalities; stochastic orderings
91B28 Finance etc. (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bäuerle, N.; Müller, A., Modeling and comparing dependencies in multivariate risk portfolios, ASTIN bulletin, 28, 59-76, (1998) · Zbl 1137.91484
[2] Brockett, P.; Garven, J., A reexamination of the relationship between preferences and moment orderings by rational risk averse investors, Geneva papers on risk and insurance theory, 23, 127-137, (1998)
[3] Denuit, M.; Lefèvre, C., Stochastic product orderings, with applications in actuarial sciences, Bulletin français d’actuariat, 1, 61-82, (1997)
[4] Denuit, M.; De Vylder, F.; Lefèvre, C., Extrema with respect to s-convex orderings in moment spaces: a general solution, Insurance: mathematics and economics, 24, 201-217, (1999)
[5] Dhaene, J.; Denuit, M., The safest dependency structure among risks, Insurance: mathematics and economics, 25, 11-21, (1999) · Zbl 1072.62651
[6] Dhaene, J.; Goovaerts, M., Dependency of risks and stop-loss order, ASTIN bulletin, 26, 201-212, (1996)
[7] Dhaene, J.; Goovaerts, M.J., On the dependency of risks in the individual life model, Insurance: mathematics and economics, 19, 243-253, (1997) · Zbl 0931.62089
[8] Dhaene, J., Wang, S., Young, V., Goovaerts, M.J., 1998. Comonotonicity and maximal stop-loss premiums. Research Report 9730. DTEW, KU Leuven, p. 13, submitted for publication.
[9] Goovaerts, M.J.; Redant, R., On the distribution of IBNR reserves, Insurance: mathematics and economics, 25, 1-9, (1999) · Zbl 0949.62087
[10] Goovaerts, M.J.; Dhaene, J.; De Schepper, A., Stochastic upper bounds for present value functions, Journal of risk and insurance theory, 67.1, 1-14, (2000)
[11] Kaas, R., How to (and how not to) compute stop-loss premiums in practice, Insurance: mathematics and economics, 13, 241-254, (1994) · Zbl 0800.62681
[12] Kaas, R., Van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of actuarial risks. Institute for Actuarial Science and Econometrics, University of Amsterdam, Amsterdam, p. 144. · Zbl 0683.62060
[13] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics and economics, 21, 219-223, (1997) · Zbl 0894.90022
[14] Rogers, L.C.G.; Shi, Z., The value of an Asian option, Journal of applied probability, 32, 1077-1088, (1995) · Zbl 0839.90013
[15] Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, New York, p. 545. · Zbl 0806.62009
[16] Simon, S.; Goovaerts, M.J.; Dhaene, J., An easy computable upper bound for the price of an arithmetic Asian option, Insurance: mathematics and economics, 26.2-3, 175-184, (2000) · Zbl 0964.91021
[17] Vyncke, D., Goovaerts, M.J., Dhaene, J., 2000. Convex upper and lower bounds for present value functions, Research Report 0025, Dept. of Applied Economics, KU Leuven, Belgium. · Zbl 0971.91030
[18] Wang, S.; Dhaene, J., Comonotonicity, correlation order and stop-loss premiums, Insurance: mathematics and economics, 22, 3, 235-243, (1998)
[19] Wang, S.; Young, V., Ordering risks: expected utility versus yaari’s dual theory of choice under risk, Insurance: mathematics and economics, 22, 145-162, (1998) · Zbl 0907.90102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.