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On the Generalized Telegraph Process with Deterministic Jumps

Antonio Di Crescenzo () and Barbara Martinucci ()
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Antonio Di Crescenzo: Università di Salerno
Barbara Martinucci: Università di Salerno

Methodology and Computing in Applied Probability, 2013, vol. 15, issue 1, 215-235

Abstract: Abstract We consider a semi-Markovian generalization of the integrated telegraph process subject to jumps. It describes a motion on the real line characterized by two alternating velocities with opposite directions, where a jump along the alternating direction occurs at each velocity reversal. We obtain the formal expressions of the forward and backward transition densities of the motion. We express them as series in the case of Erlang-distributed random times separating consecutive jumps. Furthermore, a closed form of the transition density is given for exponentially distributed times, with constant jumps and random initial velocity. In this case we also provide mean and variance of the process, and study the limiting behaviour of the probability law, which leads to a mixture of three Gaussian densities.

Keywords: Random motions; Integrated telegraph process; Alternating renewal process; Jump process; Erlang-distributed times; Asymptotic density; 60J75; 60K15 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s11009-011-9235-x

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