A stable Langevin model with diffusive-reflective boundary conditions

Stochastic Processes and their Applications. 2019. Vol. 129. No. 11. P. 4269–4293.

Jabir J. M., Profeta C.

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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.

Keywords: boundary conditions, Integrated stable Lévy processes, Hitting times, Diffusive-reflective

Publication based on the results of: Analysis of stochastic events described by the fractional stochastic equations, Langevin dynamics with elastic collisions, analysis of extremal events (2019)