Differential and integral equations for jump random motions

In this paper we obtain a differential equation for the characteristic function of random jump motion on the line, where the direction alternations and random jumps occur according to the renewal epochs of the Erlang distribution. We also study random jump motion in higher dimensions and we obtain a renewal-type equation for the characteristic function of the process. In the 3-dimensional case we obtain the telegraph-type differential equation for jump random motion, where the direction alternations and random jumps occur according to the renewal epochs of the Erlang-2 distribution.