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Finitely Additive FTAP under an Atomic Reference Measure

  • Patrizia Berti
  • Luca Pratelli
  • Pietro Rigo
Part of the Communications in Computer and Information Science book series (CCIS, volume 300)

Abstract

Let L be a linear space of real bounded random variables on the probability space \((\Omega,\mathcal{A},P_0)\). A finitely additive probability P on \(\mathcal{A}\) such that
$$ P\sim P_0 \text{ and } E_P(X)=0\text{ for each }X\in L $$

is called EMFA (equivalent martingale finitely additive probability). In this note, EMFA’s are investigated in case P 0 is atomic. Existence of EMFA’s is characterized and various examples are given. Given y ∈ ℝ and a bounded random variable Y, it is also shown that \(X_n+y\overset{a.s.}\longrightarrow Y\), for some sequence (X n ) ⊂ L, provided EMFA’s exist and E P (Y) = y for each EMFA P.

Keywords

Equivalent martingale measure Finitely additive probability Fundamental theorem of asset pricing Price uniqueness 

2000 Mathematics Subject Classification

60A05 60A10 28C05 91B25 91G10 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Patrizia Berti
    • 1
  • Luca Pratelli
    • 2
  • Pietro Rigo
    • 3
  1. 1.Dipartimento di Matematica Pura ed Applicata “G. Vitali”Università di Modena e Reggio-EmiliaModenaItaly
  2. 2.Accademia NavaleLivornoItaly
  3. 3.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly

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