  We study two-state (dichotomous, telegraph) random ergodic continuous-time
processes with dynamics depending on their past. We take into account the
history of process in an explicit form by introducing an integral non-local
memory term into the conditional probability function. We start from an
expression for the conditional transition probability function describing
additive multi-step binary random chain and show that the telegraph processes
can be considered as continuous-time interpolations of discrete-time
dichotomous random sequences. An equation involving the memory function and the
two-point correlation function of the telegraph process is analytically
obtained. This integral equation defines the correlation properties of the
processes with given memory functions. It also serves as a tool for solving the
inverse problem, namely for generation of a telegraph process with a prescribed
pair correlation function. We obtain analytically the correlation functions of
the telegraph processes with two exactly solvable examples of memory functions
and support these results by numerical simulations of the corresponding
telegraph processes.
