Sticky CIR process with potential: invariant measure and exact sampling

TL;DR

This paper analyzes the sticky CIR process with a boundary at zero, providing explicit solutions, invariant measures, and two sampling algorithms, including an exact sampler and a biased Langevin method.

Contribution

It introduces explicit Green's functions, constructs an exact sampler for the invariant measure, and develops two sampling algorithms for the sticky CIR process with potential.

Findings

The invariant measure is a mixture of a point mass and a gamma-type density.

The explicit Green's function is expressed via confluent hypergeometric functions.

The Metropolis-Hastings sampler accurately targets the invariant measure at all step sizes.

Abstract

We study the sticky Cox-Ingersoll-Ross (CIR) process in one dimension, a diffusion on $[0,\infty)$ with a sticky boundary condition at the origin, arising as the marginal process in a sparse Bayesian inference framework based on Hadamard-Langevin dynamics. For the parameter range $\delta\in(1,2)$, in which the origin is accessible but not absorbing, we prove well-posedness of the process and uniqueness of its invariant measure, which is a mixture of a point mass at zero and a weighted gamma-type density on the interior. We derive an explicit Green's function for the resolvent in terms of confluent hypergeometric functions, and use this to construct an exact sampler for the invariant measure in the zero-potential case. For a non-trivial potential $G$, we establish existence and uniqueness of the tilted invariant measure via a Girsanov change of measure, and develop two sampling algorithms: a Metropolis-Hastings corrected sampler that targets the invariant measure exactly, and an unadjusted Langevin algorithm (ULA) that is cheaper per step but introduces an $O(h)$ bias. Numerical experiments confirm the predicted behaviour: the Metropolis-Hastings sampler achieves the target invariant measure at all step sizes, while the ULA exhibits the expected $O(h)$ bias.