The Annals of Probability

A tree approach to p-variation and to integration

Jean Picard

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Abstract

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, Lévy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given).

Article information

Source
Ann. Probab., Volume 36, Number 6 (2008), 2235-2279.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1229696602

Digital Object Identifier
doi:10.1214/07-AOP388

Mathematical Reviews number (MathSciNet)
MR2478682

Zentralblatt MATH identifier
1157.60055

Subjects
Primary: 60G17: Sample path properties 60H05: Stochastic integrals 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]

Keywords
Lebesgue–Stieltjes integrals rough paths real trees variations of paths fractional Brownian motion Lévy processes

Citation

Picard, Jean. A tree approach to p -variation and to integration. Ann. Probab. 36 (2008), no. 6, 2235--2279. doi:10.1214/07-AOP388. https://projecteuclid.org/euclid.aop/1229696602


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