Quick search | Browse volumes | |

XIII: 24, 260-280, LNM 721 (1979)

**ÉMERY, Michel**

Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)

The stability theory for stochastic differential equations was developed independently by Émery (*Zeit. für W-Theorie,* **41**, 1978) and Protter (same journal, **44**, 1978). However, these results were stated in the language of convergent subsequences instead of true topological results. Here a linear topology (like convergence in probability: metrizable, complete, not locally convex) is defined on the space of semimartingales. Side results concern the Banach spaces $H^p$ and $S^p$ of semimartingales. Several useful continuity properties are proved

Comment: This topology has become a standard tool. For its main application, see the next paper 1325

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

XIV: 16, 140-147, LNM 784 (1980)

**ÉMERY, Michel**

Métrisabilité de quelques espaces de processus aléatoires (General theory of processes, Stochastic calculus)

As a sequel to the main work of 1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete

Keywords: Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

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

XV: 34, 523-525, LNM 850 (1981)

**STRICKER, Christophe**

Sur la caractérisation des semi-martingales (General theory of processes, Stochastic calculus)

This is a sequel to the preceding paper 1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam

Une topologie sur l'espace des semimartingales (General theory of processes, Stochastic calculus)

The stability theory for stochastic differential equations was developed independently by Émery (

Comment: This topology has become a standard tool. For its main application, see the next paper 1325

Keywords: Semimartingales, Spaces of 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: 16, 140-147, LNM 784 (1980)

Métrisabilité de quelques espaces de processus aléatoires (General theory of processes, Stochastic calculus)

As a sequel to the main work of 1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete

Keywords: Spaces of semimartingales

Nature: Original

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

XV: 34, 523-525, LNM 850 (1981)

Sur la caractérisation des semi-martingales (General theory of processes, Stochastic calculus)

This is a sequel to the preceding paper 1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability

Keywords: Semimartingales, Spaces of semimartingales

Nature: Original

Retrieve article from Numdam