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A stable Langevin model with diffusive-reflective boundary conditions

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  • Jabir, Jean-François
  • Profeta, Christophe

Abstract

In this note, we consider the construction of a one-dimensional stable Langevin type process confined in the upper half-plane and submitted to diffusive-reflective boundary conditions whenever the particle position hits 0. We show that two main different regimes appear according to the values of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on integrated Brownian motion. This construction further allows to obtain the exact asymptotics of the persistence probability of the integrated stable Lévy process. In addition, the paper is concluded by solving the associated trace problem in the symmetric case.

Suggested Citation

  • Jabir, Jean-François & Profeta, Christophe, 2019. "A stable Langevin model with diffusive-reflective boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4269-4293.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4269-4293
    DOI: 10.1016/j.spa.2018.11.020
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    References listed on IDEAS

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    1. Bossy, Mireille & Jabir, Jean-François, 2011. "On confined McKean Langevin processes satisfying the mean no-permeability boundary condition," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2751-2775.
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