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Telegraph-Type Processes in Higher Dimensions

In: Telegraph Processes and Option Pricing

Author

Listed:
  • Nikita Ratanov

    (Chelyabinsk State University)

  • Alexander D. Kolesnik

    (Institute of Mathematics and Computer Science)

Abstract

In recent decades, finite-velocity stochastic motions in Euclidean spaces of various dimensions have been extensively studied. This chapter provides a comprehensive survey of the most important results related to the multidimensional generalisations of the one-dimensional Goldstein-Kac telegraph process. It turns out that multidimensional finite-velocity stochastic motions are described by equations which are much more complicated than the telegraph equations. These are the so-called hyperparabolic equations, whose differential operators are composed of integer powers of the telegraph and Laplace operators. We explore stochastic motions in dimensions 2, 3, 4 and 6 in detail, and present recent results in this field, including explicit distributions of the processes in low even-dimensional Euclidean spaces.

Suggested Citation

  • Nikita Ratanov & Alexander D. Kolesnik, 2022. "Telegraph-Type Processes in Higher Dimensions," Springer Books, in: Telegraph Processes and Option Pricing, edition 2, chapter 6, pages 297-340, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-65827-7_6
    DOI: 10.1007/978-3-662-65827-7_6
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