Abstract
Let b t be Brownian motion. We show there is a unique adapted process x t which satisfies dx t = db t except when x t is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of the process. For some ranges of the parameters this is already known. We show that if a random walk close to b t is perturbed properly, its paths are close to those of x t .
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Received: 15 October 1997 / Revised version: 18 May 1998
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Davis, B. Brownian motion and random walk perturbed at extrema. Probab Theory Relat Fields 113, 501–518 (1999). https://doi.org/10.1007/s004400050215
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DOI: https://doi.org/10.1007/s004400050215