Sticky Feller diffusions

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+ 2aXtdBt I(Xt=0)dt= 1μdℓt0(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.