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Non-local solvable birth-death processes. (English) Zbl 1498.60336

Summary: In this paper, we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K25 Queueing theory (aspects of probability theory)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
60G22 Fractional processes, including fractional Brownian motion
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