  We investigate the one-dimensional telegraph random process in the presence
of an elastic boundary at the origin. This process describes a finite-velocity
random motion that alternates between two possible directions of motion
(positive or negative). When the particle hits the origin, it is either
absorbed, with probability $\alpha$, or reflected upwards, with probability
$1-\alpha$. In the case of exponentially distributed random times between
consecutive changes of direction, we obtain the distribution of the renewal
cycles and of the absorption time at the origin. This investigation is
performed both in the case of motion starting from the origin and non-zero
initial state. We also study the probability law of the process within a
renewal cycle.
