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Arbitrage and state price deflators in a general intertemporal framework

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  • Jouini, Elyes
  • Napp, Clotilde
  • Schachermayer, Walter

Abstract

In securities markets, the characterization of the absence of arbitrage by the existence of state price deflators is generally obtained through the use of the Kreps–Yan theorem.This paper deals with the validity of this theorem (see Kreps, D.M., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15–35; Yan, J.A., 1980. Caractérisation d'une classe d'ensembles convexes de L1 ou H1. Sém. de Probabilités XIV. Lecture Notes in Mathematics 784, 220–222) in a general framework. More precisely, we say that the Kreps–Yan theorem is valid for a locally convex topological space (X,?), endowed with an order structure, if for each closed convex cone C in X such that CX? and C?X+={0}, there exists a strictly positive continuous linear functional on X, whose restriction to C is non-positive.We first show that the Kreps–Yan theorem is not valid for spaces if fails to be sigma-finite.Then we prove that the Kreps–Yan theorem is valid for topological vector spaces in separating duality X,Y, provided Y satisfies both a “completeness condition” and a “Lindelöf-like condition”.We apply this result to the characterization of the no-arbitrage assumption in a general intertemporal framework.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Mathematical Economics.

Volume (Year): 41 (2005)
Issue (Month): 6 (September)
Pages: 722-734

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Handle: RePEc:eee:mateco:v:41:y:2005:i:6:p:722-734

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References

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  1. Elyès Jouini, 2001. "Arbitrage and investment opportunities," Finance and Stochastics, Springer, vol. 5(3), pages 305-325.
  2. Jouini, Elyès & Napp, Clotilde, 2001. "Arbitrage and investment opportunities," Economics Papers from University Paris Dauphine 123456789/5591, Paris Dauphine University.
  3. Freddy Delbaen, 1992. "Representing Martingale Measures When Asset Prices Are Continuous And Bounded," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 107-130.
  4. Peter Lakner, 1993. "Martingale Measures For A Class of Right-Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 43-53.
  5. Duffie, Darrell & Huang, Chi-fu, 1986. "Multiperiod security markets with differential information : Martingales and resolution times," Journal of Mathematical Economics, Elsevier, vol. 15(3), pages 283-303, June.
  6. Mas-Colell, Andreu & Zame, William R., 1991. "Equilibrium theory in infinite dimensional spaces," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 34, pages 1835-1898 Elsevier.
  7. W. Schachermayer, 1994. "Martingale Measures For Discrete-Time Processes With Infinite Horizon," Mathematical Finance, Wiley Blackwell, vol. 4(1), pages 25-55.
  8. Clark, Stephen A., 1993. "The valuation problem in arbitrage price theory," Journal of Mathematical Economics, Elsevier, vol. 22(5), pages 463-478.
  9. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
  10. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
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Cited by:
  1. Kabanov, Yuri & Lépinette-Denis, Emmanuel, 2012. "Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs," Economics Papers from University Paris Dauphine 123456789/9714, Paris Dauphine University.
  2. Martins-da-Rocha, Victor Filipe & Riedel, Frank, 2008. "On Equilibrium Prices in Continuous Time," Economics Working Papers (Ensaios Economicos da EPGE) 672, FGV/EPGE Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
  3. Teemu Pennanen, 2011. "Arbitrage and deflators in illiquid markets," Finance and Stochastics, Springer, vol. 15(1), pages 57-83, January.

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