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Sticky CIR process with potential: invariant measure and exact sampling

This paper establishes the well-posedness and characterizes the invariant measure of the one-dimensional sticky Cox-Ingersoll-Ross process with a potential, deriving an explicit Green's function and proposing both exact and approximate sampling algorithms to simulate this distribution.

Original authors: Tony Shardlow

Published 2026-05-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tony Shardlow

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a tiny particle bouncing around on a long, narrow track that starts at zero and goes to infinity. This particle is trying to find its "home" or resting place, which statisticians call an invariant measure.

In many standard models, if this particle hits the wall at zero, it either bounces off instantly (like a rubber ball) or gets stuck there forever (like a fly on flypaper). But in the real world, especially when trying to find "sparse" solutions (where many values should be exactly zero), we need something in between: a particle that can stick to the wall for a while, but eventually let go and keep moving.

This paper, by Tony Shardlow, introduces a new mathematical model for this "sticky" behavior and provides a recipe for simulating it perfectly.

Here is the breakdown of the paper's journey, using everyday analogies:

1. The Problem: The "Sticky" Wall

The paper focuses on a specific type of random movement called the Cox–Ingersoll–Ross (CIR) process. Think of this as a particle that is naturally pushed away from zero (like a spring) but also pulled back toward a center point.

The Old Way: In previous models, if the particle got too close to zero, the "spring" pushed it away so hard it could never actually touch the wall. This meant the particle could get very small, but never exactly zero.
The New Way: The author tweaks the math so the particle can reach zero. However, instead of bouncing off instantly or getting stuck forever, it gets "sticky." It lingers at zero for a random amount of time before being nudged back into the open space.
Why it matters: This is crucial for sparse Bayesian inference. In data science, "sparsity" means finding solutions where many numbers are exactly zero (like turning off unused features in a machine learning model). This sticky model naturally produces those exact zeros, creating a "spike-and-slab" distribution (a mix of a sharp spike at zero and a smooth hill elsewhere).

2. The Blueprint: The "Green's Function"

To simulate this sticky particle, you need to know exactly how it moves. The author calculates a Green's function, which is essentially a master map or a "weather forecast" for the particle.

The Analogy: Imagine you want to know where a leaf will be after a random amount of time in a windy river. The Green's function tells you the exact probability of the leaf being at any specific spot, given where it started.
The Math Magic: The author solves this map using special mathematical functions called confluent hypergeometric functions (think of them as complex, pre-calculated curves that describe the particle's behavior). Because they have an explicit formula, they don't have to guess; they can calculate the exact answer.

3. The Perfect Sampler: The "Exact Recipe"

Using this map, the author builds an Exact Sampler (Algorithm 1).

How it works: Instead of taking tiny, shaky steps (like a drunk person walking in a straight line), this algorithm jumps the particle forward in time by a random amount. It then uses the "Green's function" map to instantly calculate exactly where the particle should land next.
The Benefit: This method is exact. It doesn't approximate; it hits the target distribution perfectly, no matter how big the time jump is. It's like teleporting the particle to its correct statistical destination rather than shuffling it there.

4. Adding Complexity: The "Potential" (The Hilly Landscape)

So far, the particle moves in a flat, empty space. But real-world data often has a "landscape" or "potential" (a function $G$ ) that pulls the particle toward certain areas (like a valley) or pushes it away (like a hill).

The Challenge: When you add this landscape, the math gets messy. The "exact" teleportation trick no longer works directly because the landscape changes the rules.
The Solution 1 (The Perfect but Slow Way): The author creates a Metropolis–Hastings sampler (Algorithm 2).
- Analogy: You take a guess at where the particle should go next using the landscape, but then you hold up a "traffic light." You check the math to see if the move is fair. If it is, you go; if not, you stay put. This guarantees the result is perfectly correct, but it takes more computing power because of the "traffic light" checks.
The Solution 2 (The Fast but Slightly Flawed Way): The author also creates an Unadjusted Langevin Algorithm (ULA) (Algorithm 3).
- Analogy: This is the "fast lane." You take the guess and the move without checking the traffic light. It's much faster and cheaper per step.
- The Catch: Because you skipped the check, there is a tiny error (bias). However, the paper proves that if you make your steps smaller, this error shrinks predictably (it gets smaller in direct proportion to the step size).

5. The Proof: The Lab Results

The author tested these methods on a computer.

The Perfect Sampler: It hit the target distribution exactly, even with large steps.
The Fast Sampler: It was faster, but showed the predicted small error. As the steps got smaller, the error vanished, confirming the theory.
The Verdict: Both methods work. If you need absolute perfection, use the "traffic light" method. If you need speed and can tolerate a tiny, controllable error, use the "fast lane" method.

Summary

This paper provides a new, mathematically rigorous way to simulate a particle that can stick to zero. It solves the problem of how to generate data that is truly "sparse" (containing exact zeros) by:

Defining the rules for a "sticky" particle.
Creating a perfect map (Green's function) to move it.
Building two tools: one that is 100% accurate (but slower) and one that is very fast (with a tiny, manageable error).

This allows researchers to better model systems where "zero" is a valid and important state, rather than just a number that is very close to zero.

Problem Statement
This paper addresses the mathematical analysis and numerical sampling of the one-dimensional sticky Cox–Ingersoll–Ross (CIR) process. The process is defined on the domain $[0, \infty)$ with a sticky boundary condition at the origin. It arises as a marginal process in a sparse Bayesian inference framework based on Hadamard–Langevin dynamics, specifically designed to generate priors with genuine sparsity (point masses at zero) mixed with continuous densities, analogous to spike-and-slab priors.

The study focuses on the parameter range $\delta \in (1, 2)$ , where the origin is accessible to the diffusion but not absorbing. In this regime, the standard CIR process reflects instantaneously at zero; the "sticky" modification introduces a sojourn condition where the process spends a non-zero amount of time at the boundary, determined by a stickiness parameter $\mu > 0$ . The paper considers both the case with a zero potential ( $G \equiv 0$ ) and a non-trivial potential function $G$ .

Methodology
The analysis relies on the theory of one-dimensional diffusions, specifically utilizing scale functions, speed measures, and diffusion local time.

Well-posedness and Invariant Measure:
- The authors first establish the well-posedness of the instantaneously reflecting CIR process.
- For the sticky case, they employ an Itô–McKean time-change construction, mapping the sticky process to a time-changed reflecting CIR process.
- The invariant measure is characterized as a mixture of a point mass at zero and a weighted gamma-type density on the interior $(0, \infty)$ . This is derived from the speed measure, which includes an atom at the origin proportional to $1/\mu$ .
Exact Sampling (Zero Potential):
- The authors derive an explicit Green's function (resolvent kernel) for the reflecting CIR process in terms of confluent hypergeometric functions ( $M$ and $U$ ).
- They extend this to the sticky case by imposing a Wentzell boundary condition on the resolvent.
- Using the resolvent identity, they derive the exact transition distribution of the process at exponentially distributed times. This distribution is shown to be a three-component mixture: an atom at zero, a density on $(0, x)$ , and a density on $(x, \infty)$ .
- An exact sampling algorithm (Algorithm 1) is constructed based on this mixture structure, allowing for exact draws from the invariant measure when $G=0$ .
Sampling with Potential (Non-Zero $G$ ):
- For a non-trivial potential $G$ , the invariant measure is shown to be a Gibbsian reweighting of the zero-potential measure by the factor $e^{-\beta G}$ .
- Existence and uniqueness of this tilted invariant measure are established via a Girsanov change of measure.
- Two sampling algorithms are developed:
  - Metropolis–Hastings (MH) Sampler (Algorithm 2): Uses the exact sticky CIR resolvent as a proposal kernel combined with a clamped Euler step for the potential drift. A correction step ensures the chain targets the exact invariant measure $\pi$ .
  - Unadjusted Langevin Algorithm (ULA) (Algorithm 3): Removes the MH correction step. It combines the Euler step with the exact sticky CIR draw but targets a biased stationary distribution $\pi_h$ .

Key Results

Theoretical Characterization: The paper proves the well-posedness of the sticky CIR process for $\delta \in (1, 2)$ and uniquely characterizes its invariant measure. It identifies the specific Wentzell boundary condition satisfied by the transition density.
Exact Green's Function: An explicit form for the resolvent kernel of the sticky CIR process is derived using confluent hypergeometric functions, enabling the construction of exact samplers.
Bias Analysis: For the ULA with potential $G$ , the paper proves that the bias between the stationary distribution of the algorithm ( $\pi_h$ ) and the true invariant measure ( $\pi$ ) is of order $O(h)$ , where $h$ is the time step. This result holds under specific smoothness and growth conditions on $G$ .
Numerical Validation: Experiments confirm that the MH sampler achieves the target invariant measure exactly across various step sizes, while the ULA exhibits the predicted $O(h)$ bias, particularly in the estimation of the boundary mass (the probability of being at zero). The ULA demonstrates higher computational efficiency (ESS per second) compared to the MH sampler, though at the cost of bias.

Significance and Claims
The paper claims to provide the first rigorous theoretical foundation and practical sampling tools for the sticky CIR process in the regime where the boundary is accessible but not absorbing. Its primary contributions are:

Exact Sampling: The ability to sample exactly from the invariant measure of the sticky CIR process without potential, which serves as a building block for sparse Bayesian inference.
Algorithmic Framework: The development of a Metropolis–Hastings corrected sampler that targets the exact posterior in sparse inference problems involving Hadamard products, and an unadjusted Langevin alternative that trades exactness for computational speed.
Theoretical Rigor: The establishment of well-posedness and the derivation of the $O(h)$ bias bound for the ULA in the context of sticky boundaries, a setting where standard Langevin analysis does not directly apply due to the singular behavior at the boundary.

The authors note that the current work is restricted to the one-dimensional case. They state that extending these results to the full $d$ -dimensional Hadamard–Langevin dynamics (coupling $u$ and $v$ components) and analyzing geometric rates of convergence uniform in the step size are left for future work.