Sticky Feller diffusions

Abstract

dXt = (bXt+c)I(Xt >0) dt+ √ 2aXt dBt I(Xt =0) dt = 1 μ d` 0 t (X) where b ∈ IR and 0 < c < a are given and fixed, B is a standard Brownian motion, and `(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.