Open Access
2023 Sticky Feller diffusions
Goran Peskir, David Roodman
Author Affiliations +
Electron. J. Probab. 28: 1-28 (2023). DOI: 10.1214/23-EJP909


We consider a Feller branching diffusion process X with drift c having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1μ(0,). We show that (i) the process X can be characterised as a unique weak solution to the SDE system



where bR and 0<c<a are given and fixed, B is a standard Brownian motion, and 0(X) is a diffusion local time process of X at 0, and (ii) the transition density function of X can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of X entirely and reduces to the Mittag-Leffler function when b=0. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov forward/backward equation of X. Letting μ0 (absorption) and μ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller [6] and Molchanov [14] respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.


Dedicated to the memory of William Feller (1906–1970)


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Goran Peskir. David Roodman. "Sticky Feller diffusions." Electron. J. Probab. 28 1 - 28, 2023.


Received: 23 February 2022; Accepted: 23 January 2023; Published: 2023
First available in Project Euclid: 21 February 2023

MathSciNet: MR4550758
zbMATH: 1517.60101
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP909

Primary: 60H20 , 60J60 , 60J65
Secondary: 35C15 , 35K20 , 35K67

Keywords: Bessel process , Brownian motion , Cox-Ingersoll-Ross model , diffusion local time , Feller branching diffusion , Gamma function , Green function , Kolmogorov forward/backward equation , Kummer’s confluent hypergeometric function , Laplace transform , Mittag-Leffler function , modified Bessel function , Ornstein-Uhlenbeck process , scale function , slowly reflecting (sticky) boundary behaviour , Speed measure , sticky (Feller) boundary condition , Stochastic differential equation , Time change , transition probability density function , Tricomi’s confluent hypergeometric function , Vasicek model

Vol.28 • 2023
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