• Bernoulli
  • Volume 21, Number 1 (2015), 437-464.

Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and Lévy processes

Witold Marek Bednorz and RafaŁ Marcin Ł ochowski

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For a real càdlàg function $f$ defined on a compact interval, its truncated variation at the level $c>0$ is the infimum of total variations of functions uniformly approximating $f$ with accuracy $c/2$ and (in opposite to the total variation) is always finite. In this paper, we discuss exponential integrability and concentration properties of the truncated variation of fractional Brownian motions, diffusions and Lévy processes. We develop a special technique based on chaining approach and using it we prove Gaussian concentration of the truncated variation for certain class of diffusions. Further, we give sufficient and necessary condition for the existence of exponential moment of order $\alpha>0$ of truncated variation of Lévy process in terms of its Lévy triplet.

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Bernoulli Volume 21, Number 1 (2015), 437-464.

First available in Project Euclid: 17 March 2015

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diffusions Gaussian processes Lévy processes sample boundedness truncated variation


Bednorz, Witold Marek; Ł ochowski, RafaŁ Marcin. Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and Lévy processes. Bernoulli 21 (2015), no. 1, 437--464. doi:10.3150/13-BEJ574.

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  • [1] Bednorz, W. (2006). A theorem on majorizing measures. Ann. Probab. 34 1771–1781.
  • [2] Bednorz, W. (2006). On a Sobolev type inequality and its applications. Studia Math. 176 113–137.
  • [3] Bednorz, W. (2010). Majorizing measures on metric spaces. C. R. Math. Acad. Sci. Paris 348 75–78.
  • [4] Fernique, X. (1978). Caractérisation de processus à trajectoires majorées ou continues. In Séminaire de Probabilités XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 691–706. Berlin: Springer.
  • [5] Fernique, X. (1983). Regularité de fonctions aléatoires non gaussiennes. In Eleventh Saint Flour Probability Summer School – 1981 (Saint Flour, 1981). Lecture Notes in Math. 976 1–74. Berlin: Springer.
  • [6] Kwapień, S. and Rosiński, J. (2004). Sample Hölder continuity of stochastic processes and majorizing measures. In Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability 58 155–163. Basel: Birkhäuser.
  • [7] Kyprianou, A. E., Schoutens, W. and Willmott, P. (2005). Exotic Option Pricing and Advanced Lévy Models. Chichester: Wiley.
  • [8] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Berlin: Springer.
  • [9] Li, W.V. and Shao, Q.M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods 533–597. Handbook of Statistics 19. Amsterdam: North-Holland.
  • [10] Łochowski, R. (2011). Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift – their characteristics and applications. Stochastic Process. Appl. 121 378–393.
  • [11] Łochowski, R. (2013). On a generalisation of the Hahn–Jordan decomposition for real càdlàg functions. Colloq. Math. 132 121–138.
  • [12] Łochowski, R. and Miłoś, P. (2013). On truncated variation, upward truncated variation and downward truncated variation for diffusions. Stochastic Process. Appl. 123 446–474.
  • [13] Łochowski, R.M. (2014). Pathwise stochastic integration with respect to semimartingales. Probab. Math. Statist. To appear.
  • [14] Picard, J. (2008). A tree approach to $p$-variation and to integration. Ann. Probab. 36 2235–2279.
  • [15] Revuz, D. and Yor, M. (2005). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293 [Fundamental Principles of Mathematical Sciences]. Berlin: Springer.
  • [16] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [17] Talagrand, M. (1990). Sample boundedness of stochastic processes under increment conditions. Ann. Probab. 18 1–49.
  • [18] Talagrand, M. (2005). The Generic Chaining. Upper and Lower Bounds of Stochastic Processes. Springer Monographs in Mathematics. Berlin: Springer.