Bernoulli

  • Bernoulli
  • Volume 17, Number 1 (2011), 114-137.

On a fractional linear birth–death process

Enzo Orsingher and Federico Polito

Full-text: Open access

Abstract

In this paper, we introduce and examine a fractional linear birth–death process $N_ν(t), t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^ν(t), t>0, k≥0$. We present a subordination relationship connecting $N_ν(t), t>0$, with the classical birth–death process $N(t), t>0$, by means of the time process $T_{2ν}(t), t>0$, whose distribution is related to a time-fractional diffusion equation.

We obtain explicit formulas for the extinction probability $p_0^ν(t)$ and the state probabilities $p_k^ν(t), t>0, k≥1$, in the three relevant cases $λ>μ, λ<μ, λ=μ$ (where $λ$ and $μ$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth–death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_{\nu}(t)$ and $\operatorname{\mathbb{V}ar}N_{\nu}(t)$ are derived and analyzed.

Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 114-137.

Dates
First available in Project Euclid: 8 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1297173835

Digital Object Identifier
doi:10.3150/10-BEJ263

Mathematical Reviews number (MathSciNet)
MR2797984

Zentralblatt MATH identifier
1284.60157

Keywords
extinction probabilities fractional derivatives fractional diffusion equations generalized birth–death process iterated Brownian motion Mittag–Leffler functions

Citation

Orsingher, Enzo; Polito, Federico. On a fractional linear birth–death process. Bernoulli 17 (2011), no. 1, 114--137. doi:10.3150/10-BEJ263. https://projecteuclid.org/euclid.bj/1297173835


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