Abstract
Let (X,A) and (Y,B) be measurable spaces. Supposewe are given a probability α on A, a probability β on B and a probability μ on the product σ-field A ⊗ B. Is there a probability ν on A⊗B, with marginals α and β, such that ν ≪ μ or ν ~ μ ? Such a ν, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of ν are provided, distinguishing ν ≪ μ from ν ~ μ.
References
[1] Berti, P., L. Pratelli, and P. Rigo (2014). A unifying view on some problems in probability and statistics. Stat. Methods Appl. 23(4), 483–500. 10.1007/s10260-014-0272-9Search in Google Scholar
[2] Berti, P., L. Pratelli, and P. Rigo (2015). Two versions of the fundamental theorem of asset pricing. Electron. J. Probab. 20, 1–21. 10.1214/EJP.v20-3321Search in Google Scholar
[3] Bhaskara Rao, K. P. S. and M. Bhaskara Rao (1983). Theory of Charges. Academic Press, New York. Search in Google Scholar
[4] Folland, G. B. (1984). Real Analysis: Modern Techniques and their Applications. Wiley, New York. Search in Google Scholar
[5] Korman, J. and R. J. McCann (2015). Optimal transportation with capacity constraints. Trans. Amer. Math. Soc. 367(3), 1501– 1521. 10.1090/S0002-9947-2014-06032-7Search in Google Scholar
[6] Ramachandran, D. (1979). Perfect Measures I and II. Macmillan, New Delhi. Search in Google Scholar
[7] Ramachandran, D. (1996). Themarginal problem in arbitrary product spaces. In Distributions with fixed marginals and related topics, 260–272. Inst. Math. Statist., Hayward. 10.1214/lnms/1215452624Search in Google Scholar
[8] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist 36, 423–439. 10.1214/aoms/1177700153Search in Google Scholar
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