We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

We consider the class of real-valued stochastic processes indexed on a compact subset of R or R2 with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the two-dimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent.

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.