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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

Retrieve article from Numdam

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