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

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