A Note on the Conditional Probabilities of the Telegraph Process

We consider the telegraph process with two velocities, $a_1>a_2\in\mathbb{R}$, and two rates of reversal, $\lambda_1,\lambda_2>0$. 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.