Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 360–378 | Cite as

Computation of p-variation

  • Vygantas ButkusEmail author
  • Rimas Norvaiša


A code for computing the p-variation of a piecewise monotone function is introduced. The code is publicly available in the R environment package under the name pvar. The algorithm is based on some properties of the p-variation of a piecewise monotone function proved in this paper. The mathematical results may have their own interest.


p-variation computation piecewise monotone function 




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  1. 1.
  2. 2.
    G. De Marco, Representing the algebra of regulated functions as an algebra of continuous functions, Rend. Semin. Mat. Univ. Padova, 84:195–199, 1990.Google Scholar
  3. 3.
    B.K. Driver, Rough Path Analysis, 2013, available from:
  4. 4.
    R.M. Dudley and R. Norvaiša, An Introduction to p-Variation and Young Integrals, with Emphasis on Sample Functions of Stochastic Processes, MaPhySto Lect. Notes, Vol. 1, Univ. Aarhus, Denmark, 1998.Google Scholar
  5. 5.
    R.M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and p-variation, Lect. Notes Math., Vol. 1703, Springer, Berlin, Heidelberg, 1999.Google Scholar
  6. 6.
    R.M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer, New York, 2010.zbMATHGoogle Scholar
  7. 7.
    P.K. Friz and N.B. Victoir, Multidimensional Stochastic Processes as Rough Paths, Cambridge Univ. Press, Cambridge, 2010.CrossRefGoogle Scholar
  8. 8.
    T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam., 14(2):215–310, 1998.MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Norvaiša, Rough functions: p-variation, calculus, and index estimation, Lith. Math. J., 46(1):102–128, 2006.MathSciNetCrossRefGoogle Scholar
  10. 10.
    R. Norvaiša and A. Račkauskas, Convergence in law of partial sum processes in p-variation norm, Lith. Math. J., 48(2):212–227, 2008.MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Norvaiša and D.M. Salopek, Estimating the p-variation index of a sample function: An application to financial data set, Methodol. Comput. Appl. Probab., 4(1):27–53, 2002.MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Qian, The p-Variation of Partial Sum Processes and the Empirical Process, PhD thesis, Tufts University, 1997.Google Scholar
  13. 13.
    J. Qian, The p-variation of partial sum processes and the empirical process, Ann. Probab., 26(3):1370–1383, 1998.MathSciNetCrossRefGoogle Scholar
  14. 14.
    R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Viena, Austria, 2011, available from:
  15. 15.
    N. Wiener, The quadratic variation of a function and its Fourier coefficients, J. Math. Phys., Mass. Inst. Techn., 3(2):72–94, 1924.zbMATHGoogle Scholar
  16. 16.
    L.C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67:251–282, 1936.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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