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Stability of stochastic differential equations in manifolds

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Stability of stochastic differential equations in manifolds SÉMINAIRE DE PROBABILITÉS (STRASBOURG) MARC ARNAUDON ANTON THALMAIER Stability of stochastic differential equations in manifolds Séminaire de probabilités (Strasbourg), tome 32 (1998), p. 188-214. <http://www.numdam.org/item?id=SPS_1998__32__188_0> © Springer-Verlag, Berlin Heidelberg New York, 1998, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Stability of stochastic differential equations in manifolds MARC ARNAUDON Institut de Recherche Mathématique Avancée Université Louis Pasteur et CNRS 7, rue René Descartes . F-67084 Strasbourg Cedex France [email protected] ANTON THALMAIER Naturwissenschaftliche Fakultät I - Mathematik Universitat Regensburg D-93040 Regensburg Germany [email protected] Abstract. - We extend the so-called topology of semimartingales to continuous semimartingales with values in a manifold and with lifetime, and prove that if the manifold is endowed with a connection V then this topology and the topology of compact convergence in probability coincide on the set of continuous V-martingales. For the topology of manifold-valued semimartingales, we give results on differentia- tion with respect to a parameter for second order, Stratonovich and Itô stochastic differential equations and identify the equation solved by the derivative processes. In particular, we prove that both Stratonovich and Itô equations differentiate like equations involving smooth pa

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