VI: 03, 51-71, LNM 258 (1972)
BRETAGNOLLE, Jean
$p$-variation de fonctions aléatoires~: 1. Séries de Rademacher 2. Processus à accoissements indépendants (
Independent increments)
The main result of the paper is theorem III, which gives a necessary and sufficient condition for the sample paths of a centered Lévy process to have a.s. a finite $p$-variation on finite time intervals, for $1<p<2$: the process should have no Gaussian part, and $|x|^p$ be integrable near $0$ w.r.t. the Lévy measure $L(dx)$. The proof rests on discrete estimates on the $p$-variation of Rademacher series. Additional results on $h$-variation w.r.t. more general convex functions are given or mentioned
Comment: This paper improves on Millar,
Zeit. für W-theorie, 17, 1971
Keywords: $p$-variation,
Rademacher functionsNature: Original
Retrieve article from Numdam
XIII: 18, 227-232, LNM 721 (1979)
BRUNEAU, Michel
Sur la $p$-variation d'une surmartingale continue (
Martingale theory)
The $p$-variation of a deterministic function being defined in the obvious way as a supremum over all partitions, the sample functions of a continuous martingale (and therefore semimartingale) are known to be of finite $p$-variation for $p>2$ (not for $p=2$ in general: non-anticipating partitions are not sufficient to compute the $p$-variation). If $X$ is a continuous supermartingale, a universal bound is given on the expected $p$-variation of $X$ on the interval $[0,T_\lambda]$, where $T_\lambda=\inf\{t:|X_t-X_0|\ge\lambda\}$. The main tool is Doob's classical upcrossing inequality
Comment: For an extension see
1319. These properties are used in T.~Lyons' pathwise theory of stochastic differential equations; see his long article in
Rev. Math. Iberoamericana 14, 1998
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 19, 233-237, LNM 721 (1979)
STRICKER, Christophe
Sur la $p$-variation des surmartingales (
Martingale theory)
The method of the preceding paper of Bruneau
1318 is extended to all right-continuous semimartingales
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 20, 238-239, LNM 721 (1979)
STRICKER, Christophe
Une remarque sur l'exposé précédent (
Martingale theory)
A few comments are added to the preceding paper
1319, concerning in particular its relationship with results of Lépingle,
Zeit. für W-Theorie, 36, 1976
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XLIII: 08, 215-219, LNM 2006 (2011)
PRATELLI, Maurizio
A Remark on the $1/H$-variation of the Fractional Brownian Motion (
Theory of processes)
Keywords: Fractional Brownian motion,
$p$-variation,
Ergodic theoremNature: Exposition
XLIII: 11, 269-307, LNM 2006 (2011)
PAGÈS, Gilles;
SELLAMI, Afef
Convergence of multi-dimensional quantized SDE's (
Integration theory,
Theory of processes)
Keywords: Functional quantization,
Stochastic differential equations,
Stratonovich integrals,
Stationary quantizers,
Rough paths,
Itô map,
Hölder semi-norm,
$p$-variationNature: Original
