Functionals of Telegraph Process

Nikita Ratanov and Alexander Kolesnik
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Nikita Ratanov: Chelyabinsk State University
Alexander Kolesnik: Institute of Mathematics and Computer Science

Chapter 5 in Telegraph Processes and Option Pricing, 2022, pp 223-296 from Springer

Abstract: Abstract This chapter is devoted to the various functionals of telegraph processes. First, we present well-known results on distributions of the telegraph processes distorted by reflecting and absorbing barriers. The next group of results give a detailed description of the share of the time spent by the telegraph process on the positive semi-axis. These results generalise Paul Lévy’s famous and beautiful Arcsine Law, which dates back about 80 years. The distribution of exponential functional based on the asymmetric telegraph is also introduced. Problems of the first passage time are then studied in various settings, for a continuous telegraph process as well as for processes with alternating random jumps, and we look at the distribution of telegraphic meanders and running extrema. The chapter ends with recent results on the distribution of distance between two telegraph processes.

Date: 2022
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DOI: 10.1007/978-3-662-65827-7_5

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