## 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

where b ∈ R and 0 < c < a are given and fixed, B is a standard Brownian motion,

and l

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.

*dX*

I(X_{t }= (bX_{t}+c) I (X_{t }>0) dt+√2aX_{t}dB_{t}I(X

_{t}= 0) dt = 1/µ dl^{0}_{t }(X)where b ∈ R and 0 < c < a are given and fixed, B is a standard Brownian motion,

and l

^{0}(X) is a diffusion local time process of X at 0, and (ii) the transition densityfunction 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 language | English |
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Journal | Electronic Journal of Probability |

DOIs | |

Publication status | Published - 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