A Note on Higher Dimensional p-Variation

Peter Friz (Technische Universit├Ąt and Weierstrass Institute for Applied Analysis and Stochastics, Berlin)
Nicolas Victoir (New York)


We discuss $p$-variation regularity of real-valued functions defined on $[0,T]\times [0,T]$, based on rectangular increments. When $p>1$, there are two slightly different notions of $p$-variation; both of which are useful in the context of Gaussian roug paths. Unfortunately, these concepts were blurred in previous works; the purpose of this note is to show that the afore-mentioned notions of $p$-variations are "epsilon-close". In particular, all arguments relevant for Gaussian rough paths go through with minor notational changes.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1880-1899

Publication Date: October 16, 2011

DOI: 10.1214/EJP.v16-951


  1. L. Coutin, Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motion Prob. Th. Rel. Fields 122 (2002), 108-140. MR1883719 (2003c:60066)
  2. P. Friz, N. Victoir. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincare, Probab. Stat. 46 no. 2 (2010), no. 2, 369-41.
  3. P. Friz, N. Victoir. Multidimensional stochastic processes as rough paths. Theory and applications. Cambridge Studies of Advanced Mathematics, Vol. 120, Cambridge University Press, 670 pages. Errata available here
  4. N. Towghi: Multidimensional extension of L.C.Young's Inequality, J. Inequal. Pure Appl. Math., Vol 3, 2-22, (2002), 13 pp. (electronic), MR1906391(2003c:26035)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.