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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 functions

Nature: Original

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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, Upcrossings

Nature: Original

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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, Upcrossings

Nature: Original

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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, Upcrossings

Nature: Original

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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 theorem

Nature: 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, $p$-variation

Nature: Original

$p$-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,

Keywords: $p$-variation, Rademacher functions

Nature: Original

Retrieve article from Numdam

XIII: 18, 227-232, LNM 721 (1979)

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

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 19, 233-237, LNM 721 (1979)

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, Upcrossings

Nature: Original

Retrieve article from Numdam

XIII: 20, 238-239, LNM 721 (1979)

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,

Keywords: $p$-variation, Upcrossings

Nature: Original

Retrieve article from Numdam

XLIII: 08, 215-219, LNM 2006 (2011)

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIII: 11, 269-307, LNM 2006 (2011)

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, $p$-variation

Nature: Original