Abstract
We consider the telegraph process with two velocities, a1>a2∈R, and two rates of reversal, λ1,λ2>0. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time t>0 conditioned on its position at a previous time 0<s<t. In the case of a symmetric process, we present some results on the joint distribution of the position of the motion at time t>0, its maximum and its minimum up to that moment.
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