Sticky CIR process with potential: invariant measure and exact sampling

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 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 a cheaper, biased unadjusted Langevin algorithm (ULA) for a boundary-clamped variant, for which we prove a first-order expansion of the stationary bias with an explicit constant, its leading term of order $h |\log h |$, independent of $\delta$. Numerical experiments confirm the predicted behaviour: the Metropolis--Hastings sampler achieves the target invariant measure at all step sizes, while the ULA bias follows the proven first-order law, including its constant.