Non-Local Solvable Birth-Death Processes

Abstract

In this paper we study strong solutions of some non-local differencedifferential 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.

Non-Local Pearson diffusions

“…admits a unique solution for any λ > 0 and it is given by e Φ (t; λ) := E[e λEΦ(t) ] (hence, in particular, it is a completely monotone function in λ for fixed t). In [6] the following proposition is proved. Proposition 3.1.…”

“…However, since the process X Φ (t) (under the hypothesis that X Φ (0) admits m(x)dx as distribution) is first-order stationary, but not second-order stationary (neither in wide sense), we cannot exploit long-range or short-range dependence properties with the usual definitions. Thus, as we did in [6], following the lines of [12, Lemmas 2.1 and 2.2], we give the following definitions.…”

“…Indeed, for instance, in [58,17] a link between abstract generalized fractional differential equations and time-changed semigroups is established, while in [19] properties of the Green measures of time-changed Markov properties are exploited. Moreover, such non-local derivatives have been used to define a class of non-local birth-death processes (see [6]) whose stationary distributions fall into the Ord family (in particular the Katz family), which are discrete analogous (and actual lattice approximations) of Pearson diffusions of the first spectral category. On the other hand, a first generalization to the non-local setting of a Pearson diffusion (in particular Ornstein-Uhlenbeck processes) has been achieved in [22].…”