Approximation via regularization of the local time of semimartingales and Brownian motion

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Niveau: Supérieur, Licence, Bac+2
Approximation via regularization of the local time of semimartingales and Brownian motion Berard Bergery Blandine1, Vallois Pierre1 (1) Universite Henri Poincare, Institut de Mathematiques Elie Cartan, B.P. 239, F-54506 Vandœuvre-les- Nancy Cedex, France Abstract: Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartin- gales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in L2, and a.s. convergence of some of our schemes. Keywords: local time, stochastic integration by regularization, quadratic variation, rate of con- vergence, stochastic Fubini's theorem. 2000 MSC: 60G44, 60H05, 60H99, 60J55, 60J60, 60J65. 1 Introduction Let (Xt)t>0 be a continuous process which is defined on a complete probability space (?,F ,Ft, P ). It is supposed that (Ft) verifies the usual hypotheses. 1. Let Xt = Mt + Vt be a continuous (Ft)-semimartingale, where M is a local martingale and V is an adapted process with finite variation. In the usual stochastic calculus, two fundamental processes are associated with X: its quadratic variation and its local time.

  • admits local

  • let

  • stochastic integration

  • continuous semimartin- gales

  • time process

  • local time

  • holder continuity

  • negative real


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INSTITUT NATIONAL DE LA STATISTIQUE ET DES ETUDES ECONOMIQUES
Série des Documents de Travail du CREST
(Centre de Recherche en Economie et Statistique)










n° 2007-24

The Econometrics of Auctions
with Asymmetric
*Anonymous Bidders

1L. LAMY





























Les documents de travail ne reflètent pas la position de l'INSEE et n'engagent que
leurs auteurs.

Working papers do not reflect the position of INSEE but only the views of the authors.

* I would like to thank Philippe Février and Bernard Sanalié for very helpful discussions.
1 Laboratoire d’Economie Industrielle, CREST-INSEE, 28 Rue des Saints-Pères, 75007 PARIS, France.
laurent.lamy@ensae.fr ∗



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