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Abstract : |
The paper deals with one-dimensional Brownian motion perturbed when it hits its minimum and/or its maximum. It first presents some features of perturbed reflected Brownian motion defined as jBj \Gamma ` where B is standard Brownian motion, ` its local time at 0 and ? ? a positive constant; in particular it is shown that the positive and negative excursions in the sense of Ito of this "one--sided " perturbed Brownian motion are two independent point processes, which in turn is used to construct two--sided perturbed Brownian motion. It is then shown that the constructed process is almost surely the unique solution of the implicit stochastic equation studied by Le Gall [11], Carmona, Petit and Yor [5] and Davis [6], even when no restrictions are imposed on the partial reflections. Finally, the Hausdorff dimension of sets of exceptional points for perturbed Brownian motion such as points of monotonicity are computed. Mathematics Subject Classification (1991): 60J65, |
