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XIV: 12, 116-117, LNM 784 (1980)

**CHOU, Ching Sung**

Une caractérisation des semimartingales spéciales (Stochastic calculus)

This is a useful addition to the next paper 1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special

Comment: See also 1352

Keywords: Semimartingales, Métivier-Pellaumail inequality, Special semimartingales

Nature: Original

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XIV: 13, 118-124, LNM 784 (1980)

**ÉMERY, Michel**

Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (Stochastic calculus)

Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see 1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales

Comment: See 1352. A general reference on the Métivier-Pellaumail method can be found in their book*Stochastic Integration,* Academic Press 1980. See also He-Wang-Yan, *Semimartingale Theory and Stochastic Calculus,* CRC Press 1992

Keywords: Semimartingales, Spaces of semimartingales, Stochastic differential equations, Doob's inequality, Métivier-Pellaumail inequality

Nature: Original

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XIV: 14, 125-127, LNM 784 (1980)

**LENGLART, Érik**

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

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XIV: 24, 209-219, LNM 784 (1980)

**PELLAUMAIL, Jean**

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail*Stochastic Integration* (1980), the class of semimartingales being defined by the Métivier-Pellaumail inequality (1413)

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

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XV: 33, 499-522, LNM 850 (1981)

**STRICKER, Christophe**

Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (Stochastic calculus)

This paper contains a number of useful technical results on the topology of semimartingales (see 1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see 1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see 1412, 1413, 1414)

Keywords: Semimartingales, Stochastic integrals, Spaces of semimartingales, Métivier-Pellaumail inequality

Nature: Original

Retrieve article from Numdam

Une caractérisation des semimartingales spéciales (Stochastic calculus)

This is a useful addition to the next paper 1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special

Comment: See also 1352

Keywords: Semimartingales, Métivier-Pellaumail inequality, Special semimartingales

Nature: Original

Retrieve article from Numdam

XIV: 13, 118-124, LNM 784 (1980)

Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (Stochastic calculus)

Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see 1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales

Comment: See 1352. A general reference on the Métivier-Pellaumail method can be found in their book

Keywords: Semimartingales, Spaces of semimartingales, Stochastic differential equations, Doob's inequality, Métivier-Pellaumail inequality

Nature: Original

Retrieve article from Numdam

XIV: 14, 125-127, LNM 784 (1980)

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

Retrieve article from Numdam

XIV: 24, 209-219, LNM 784 (1980)

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

Retrieve article from Numdam

XV: 33, 499-522, LNM 850 (1981)

Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (Stochastic calculus)

This paper contains a number of useful technical results on the topology of semimartingales (see 1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see 1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see 1412, 1413, 1414)

Keywords: Semimartingales, Stochastic integrals, Spaces of semimartingales, Métivier-Pellaumail inequality

Nature: Original

Retrieve article from Numdam