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 $1∕\mathrm{\mu }\in (0,\mathrm{\infty })$. We show that (i) the process X can be characterised as a unique weak solution to the SDE system

$$d{X}_t=(b{X}_t+c)I(X_t>0)dt+\sqrt{2a{X}_t}d{B}_t$$

$$I(X_t=0)dt=\frac{1}{\mathrm{\mu }}d{\ell }_t^0(X)$$

where $b\in \mathbb{R}$ and $0<c<a$ are given and fixed, B is a standard Brownian motion, and ${\ell }^0(X)$ 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 $\mathrm{\mu }\downarrow 0$ (absorption) and $\mathrm{\mu }\uparrow \mathrm{\infty }$ (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.