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
1325Keywords: Semimartingales,
Spaces of semimartingalesNature: 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 inequalityNature: 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 semimartingalesNature: 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 inequalityNature: 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 semimartingalesNature: Original Retrieve article from Numdam
