In this note, we consider the construction of a one-dimensional stable
Langevin type process confined in the upper half-plane and submitted to
reflective-diffusive 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
L{\'e}vy process. In addition, the paper is concluded by solving the associated
trace problem in the symmetric case