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Statistical Methods & Applications

, Volume 23, Issue 4, pp 483–500 | Cite as

A unifying view on some problems in probability and statistics

  • Patrizia Berti
  • Luca Pratelli
  • Pietro Rigo
Article

Abstract

Let \(L\) be a linear space of real random variables on the measurable space \((\varOmega ,\mathcal {A})\). Conditions for the existence of a probability \(P\) on \(\mathcal {A}\) such that \(E_P|X|<\infty \) and \(E_P(X)=0\) for all \(X\in L\) are provided. Such a \(P\) may be finitely additive or \(\sigma \)-additive, depending on the problem at hand, and may also be requested to satisfy \(P\sim P_0\) or \(P\ll P_0\) where \(P_0\) is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability \(P\) as above. Among them, we mention de Finetti’s coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions.

Keywords

Compatibility of conditional distributions De Finetti’s coherence principle Equivalent martingale measure Equivalent probability measure with given marginals Finitely additive probability Fundamental theorem of asset pricing 

References

  1. Berti P, Rigo P (1996) On the existence of inferences which are consistent with a given model. Ann Stat 24:1235–1249CrossRefzbMATHMathSciNetGoogle Scholar
  2. Berti P, Rigo P (1999) Sufficient conditions for the existence of disintegrations. J Theor Probab 12:75–86CrossRefzbMATHMathSciNetGoogle Scholar
  3. Berti P, Pratelli L, Rigo P (2013) Finitely additive equivalent martingale measures. J Theor Probab 26:46–57CrossRefzbMATHMathSciNetGoogle Scholar
  4. Berti P, Pratelli L, Rigo P (2014a) Price uniqueness and fundamental theorem of asset pricing with finitely additive probabilities. Stochastics 86:135–146zbMATHMathSciNetGoogle Scholar
  5. Berti P, Pratelli L, Rigo P (2014b) Two versions of the fundamental theorem of asset pricing (submitted). http://www-dimat.unipv.it/~rigo/
  6. Berti P, Dreassi E, Rigo P (2014) Compatibility results for conditional distributions. J Multivar Anal 125:190–203CrossRefzbMATHMathSciNetGoogle Scholar
  7. Cassese G (2007) Yan theorem in \(L_\infty \) with applications to asset pricing. Acta Math Appl Sin Engl Ser 23:551–562CrossRefzbMATHMathSciNetGoogle Scholar
  8. Dalang R, Morton A, Willinger W (1990) Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch Stoch Rep 29:185–201CrossRefzbMATHMathSciNetGoogle Scholar
  9. Delbaen F, Schachermayer W (2006) The mathematics of arbitrage. Springer, BerlinzbMATHGoogle Scholar
  10. Dubins LE, Prikry K (1995) On the existence of disintegrations. Seminaire de Probabilités XXIX. Lecture Notes in Mathematics. Springer, Berlin. vol 1613, pp 248–259Google Scholar
  11. Heath D, Sudderth WD (1978) On finitely additive priors, coherence and extended admissibility. Ann Stat 6:333–345CrossRefzbMATHMathSciNetGoogle Scholar
  12. Rokhlin DB (2005) The Kreps-Yan theorem for \(L^\infty \). Intern J Math Math Sci 17:2749–2756CrossRefMathSciNetGoogle Scholar
  13. Rokhlin DB, Schachermayer W (2006) A note on lower bounds of martingale measure densities. Ill J Math 50:815–824zbMATHMathSciNetGoogle Scholar
  14. Strassen V (1965) The existence of probability measures with given marginals. Ann. Math. Stat. 36:423–439CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed Applicata “G. Vitali”Universita’ di Modena e Reggio-EmiliaModenaItaly
  2. 2.Accademia NavaleLivornoItaly
  3. 3.Dipartimento di Matematica “F. Casorati”Universita’ di PaviaPaviaItaly

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