Acta Applicandae Mathematicae

, Volume 138, Issue 1, pp 1–15 | Cite as

Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

Article

Abstract

Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in (Mbele Bidima and Rásonyi in Ann. Oper. Res. 200:131–146, 2012), we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to 1 geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory.

Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper.

Keywords

Asymptotic exponential arbitrage Markov chains Large deviations Expected utility 

References

  1. 1.
    Bhattacharya, R.N., Waymire, E.C.: Stochastic Processes with Applications. Wiley, New York (1990) Google Scholar
  2. 2.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  3. 3.
    Dokuchaev, N.: Mean-reverting market models: speculative opportunities and non-arbitrage. Appl. Math. Finance 14, 319–337 (2007) MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Du, K., Neufeld, A.D.: A note on asymptotic exponential arbitrage with exponentially decaying failure probability. J. Appl. Probab. 50, 801–809 (2013) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Föllmer, H., Schachermayer, W.: Asymptotic arbitrage and large deviations. Math. Financ. Econ. 1, 213–249 (2007) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter, Berlin (2004) CrossRefGoogle Scholar
  7. 7.
    Kabanov, Y.M., Kramkov, D.O.: Asymptotic arbitrage in large financial markets. Finance Stoch. 2, 143–172 (1998) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kontoyiannis, I., Meyn, S.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13, 304–362 (2003) MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kontoyiannis, I., Meyn, S.: Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10, 61–123 (2005) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mbele Bidima, M.L.D., Rásonyi, M.: On long-term arbitrage opportunities in Markovian models of financial markets. Ann. Oper. Res. 200, 131–146 (2012) MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Martin Le Doux Mbele Bidima
    • 1
  • Miklós Rásonyi
    • 2
    • 3
  1. 1.University of Yaoundé IYaoundéCameroon
  2. 2.MTA Alfréd Rényi Institute of MathematicsBudapestHungary
  3. 3.University of EdinburghEdinburghUK

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