Minimal \(\mathcal{L}^p \)-densities with prescribed marginals. (English) Zbl 1473.46037

Summary: We derive sharp lower bounds for \(\mathcal{L}^p \)-functions on the \(n\)-dimensional unit hypercube in terms of their \(p\)-th marginal moments. Such bounds are the unique solutions of a system of constrained nonlinear integral equations depending on the marginals. For square-integrable functions, the bounds have an explicit expression in terms of the second marginals’ moments.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60E15 Inequalities; stochastic orderings
45G15 Systems of nonlinear integral equations
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI arXiv Euclid


[1] Breeden, D.T. and Litzenberger, R.H. (1978). Prices of state-contingent claims implicit in option prices. J. Bus. 621-651.
[2] Carr, P. and Madan, D. (2001). Towards a theory of volatility trading. In Option Pricing, Interest Rates and Risk Management. Handb. Math. Finance 458-476. Cambridge: Cambridge Univ. Press. Zentralblatt MATH: 0990.91037
· Zbl 0990.91037
[3] Green, R.C. and Jarrow, R.A. (1987). Spanning and completeness in markets with contingent claims. J. Econom. Theory 41 202-210. Zentralblatt MATH: 0621.90015
Digital Object Identifier: doi:10.1016/0022-0531(87)90014-7
· Zbl 0621.90015
[4] Guasoni, P. and Mayerhofer, E. (2020). Technical note – options portfolio selection. Oper. Res. 68 733-740. Zentralblatt MATH: 07269924
Digital Object Identifier: doi:10.1287/opre.2019.1925
· Zbl 1455.91235
[5] Megginson, R.E. (2012). An Introduction to Banach Space Theory 183. Berlin: Springer.
[6] Nachman, D.C. (1988). Spanning and completeness with options. Rev. Financ. Stud. 1 311-328.
[7] Nelsen, R.B. (2007). An Introduction to Copulas. Berlin: Springer. Zentralblatt MATH: 1152.62030
· Zbl 1152.62030
[8] Rudin, W. (2006). Real and Complex Analysis. New York: McGraw-Hill. Zentralblatt MATH: 0925.00005
· Zbl 0925.00005
[9] Shapiro, H.
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.