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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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