Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 141–156 | Cite as

On the quadratic variation of the model-free price paths with jumps

  • Lesiba Charles Galane
  • Rafał Marcin Łochowski
  • Farai Julius Mhlanga


We prove that the model-free typical (in the sense of Vovk) càdlàg price paths with mildly restricted downward jumps possess quadratic variation, which does not depend on the specific sequence of partitions as long as these partitions are obtained from stopping times such that the oscillations of a path on the consecutive (half-open on the right) intervals of these partitions tend (in a specified sense) to 0. Finally, we also define quasi-explicit, partition-independent quantities that tend to this quadratic variation.


Vovk’s outer measure càdlàg price paths Lebesque partition quadratic variation truncated variation 


60H05 91G99 


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  1. 1.
    M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21(1):360–373, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Cont and D.A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259:1043–1072, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Davis, J. Obłój, and P. Siorpaes, Pathwise stochastic calculus with local times, Ann. Inst. H. Poincaré Probab. Stat., 54(1):1–21, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H. Föllmer, Calcul d’Itô sans probabilités, in J. Azéma and M. Yor (Eds.), Séminaire de Probabilités XV, Univ. Strasbourg 1979/80, Lect. Notes Math., Vol. 850, Springer, Berlin, Heidelberg, 1981, pp. 143–150.Google Scholar
  5. 5.
    R. Łochowski, Quadratic variation of càdlàg semimartingales as a.s. limit of the normalized truncated variations, Stochastics, 2018 (to appear), arXiv:1708.00732.Google Scholar
  6. 6.
    R. Łochowski, N. Perkowski, and D. Prömel, A superhedging approach to stochastic integration, Stoch. Processes Appl., 2018, available from:
  7. 7.
    R.M. Łochowski, On pathwise stochastic integration with respect to semimartingales, Probab.Math. Stat., 34(1):23–43, 2014.MathSciNetzbMATHGoogle Scholar
  8. 8.
    R.M. Łochowski and P. Miłoś, On truncated variation, upward truncated variation and downward truncated variation for diffusions, Stoch. Processes Appl., 123(2):446–474, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. Perkowski and D.J. Prömel, Pathwise stochastic integrals for model free finance, Bernoulli, 22(4):2486–2520, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H.M. Taylor, A stopped Brownian motion formula, Ann. Probab., 2(3):234–246, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    V. Vovk, Continuous-time trading and the emergence of probability, Finance Stoch., 16(4):561–609, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. Vovk, Itô calculus without probability in idealized financial markets, Lith. Math. J., 55(2):270–290, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    V. Vovk, Purely pathwise probability-free Itô integral, Mat. Stud., 46(2):96–110, 2016.MathSciNetzbMATHGoogle Scholar
  14. 14.
    V. Vovk and G. Shafer, Towards a probability-free theory of continuous martingales, 2017 (submitted for publication), arXiv:1703.08715.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Lesiba Charles Galane
    • 1
  • Rafał Marcin Łochowski
    • 2
  • Farai Julius Mhlanga
    • 1
  1. 1.Department of Mathematics and Applied MathematicsUniversity of LimpopoSovengaSouth Africa
  2. 2.Department of Mathematics and Mathematical Economics, Warsaw School of EconomicsWarszawaPoland

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