Sticky Feller diffusions

Goran Peskir, David Roodman

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a Feller branching diffusion process X with drift c having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1/µ ∈ (0, ∞). We show that (i) the process X can be characterised as a unique weak solution to the SDE system
dXt = (bXt+c) I (Xt >0) dt+√2aXt dBt
I(Xt = 0) dt = 1/µ dl
0t (X)
where b ∈ R and 0 < c < a are given and fixed, B is a standard Brownian motion,
and l0(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 µ ↓ 0 (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.
Original languageEnglish
JournalElectronic Journal of Probability
DOIs
Publication statusPublished - 21 Feb 2023

Keywords

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

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