Square-Root Diffusions and the Feller Condition

Why the Boundary Matters

A concentration model with random fluctuations must say what happens near zero. A Gaussian diffusion can cross below zero, while a square-root diffusion reduces its random fluctuations as the state approaches zero. That reduction alone does not guarantee that zero cannot be reached: the balance between the inward drift and the noise level also matters.

The Cox–Ingersoll–Ross, or CIR, diffusion makes this balance explicit. Its Feller condition is a parameter inequality that determines whether a process starting at a positive concentration can hit zero. The condition is about the continuous-time model itself, not about whether a particular numerical path happens to remain positive.

Definition: The CIR square-root diffusion is the nonnegative solution of $$ dX_t=\kappa(\theta-X_t)\,dt+\sigma\sqrt{X_t}\,dW_t, $$ where $W_t$ is standard Brownian motion, $\kappa>0$ is the mean-reversion rate, $\theta\geq 0$ is the long-run mean level, and $\sigma>0$ is the diffusion scale. The initial state is $X_0=x>0$, unless stated otherwise. The Feller condition is $2\kappa\theta\geq\sigma^2$.

The drift pulls the state toward $\theta$: it is positive below $\theta$ and negative above it. The diffusion coefficient is $\sigma\sqrt{X_t}$, so the instantaneous variance rate is $\sigma^2X_t$. At zero the diffusion coefficient vanishes, while the drift is $\kappa\theta$. The Feller condition quantifies whether this boundary drift is strong enough, relative to the noise, to make zero inaccessible from the interior.

For a concentration-valued $X_t$, $\kappa$ has units of inverse time, $\theta$ has concentration units, and $\sigma$ has units of square-root-of-concentration per square-root-time. Thus $\sigma\sqrt{X_t}\,dW_t$ has concentration units, and both $2\kappa\theta$ and $\sigma^2$ have concentration-per-time units. Their ratio is dimensionless.

Scale and the Boundary Test

The generator of this one-dimensional diffusion, acting on twice differentiable functions $f$, is $$ \mathcal{L}f(x)=\kappa(\theta-x)f'(x)+\frac{1}{2}\sigma^2x f''(x),\qquad x>0. $$ For a diffusion with drift $b(x)$ and variance rate $a(x)>0$, a scale density is $$ s'(x)=\exp\left(-\int^x \frac{2b(y)}{a(y)}\,dy\right). $$ It gives a scale function $s(x)=\int^x s'(y)\,dy$, up to an arbitrary additive constant. The speed density, with this convention, is $m(x)=2/(a(x)s'(x))$. Multiplying $s'$ by a positive constant changes the speed density by the reciprocal constant, but does not change the boundary classification.

Worked Example: Deriving the CIR Scale Density

For the CIR model, $b(x)=\kappa(\theta-x)$ and $a(x)=\sigma^2x$. Define $\alpha=2\kappa\theta/\sigma^2$ and $\beta=2\kappa/\sigma^2$. Then $$ \frac{2b(x)}{a(x)} =\frac{2\kappa(\theta-x)}{\sigma^2x} =\frac{\alpha}{x}-\beta. $$

Integrating gives $\int(2b/a)\,dx=\alpha\log x-\beta x+C$. Substitution in the scale-density definition, and omission of an irrelevant positive multiplicative constant, yields $$ s'(x)=x^{-\alpha}e^{\beta x}. $$ The corresponding speed density is $$ m(x)=\frac{2}{\sigma^2x\,s'(x)} =\frac{2}{\sigma^2}x^{\alpha-1}e^{-\beta x}. $$ These expressions expose the behavior near zero: the exponential factors tend to one there, so the power $x^{-\alpha}$ in the scale density determines whether the scale integral diverges.

The relevant one-dimensional diffusion boundary test is Feller’s boundary classification. In the scale-and-speed convention above, zero is inaccessible from the interior if $s(0+)=-\infty$. If $s(0+)$ is finite, zero is accessible when $$ \int_0^c [s(x)-s(0+)]m(x)\,dx<\infty $$ for some interior $c>0$. “Accessible” means that the process can reach the boundary from the interior with positive probability. The test is applied to the diffusion on $(0,\infty)$, with the coefficients locally regular there; the CIR coefficients satisfy these conditions on every compact subinterval of $(0,\infty)$.

Theorem: For the CIR diffusion with $\kappa>0$, $\theta\geq0$, and $\sigma>0$, zero is inaccessible from any $x>0$ if and only if $2\kappa\theta\geq\sigma^2$. If $2\kappa\theta<\sigma^2$, zero is accessible.

Determine scale behavior.
Near zero, $s'(x)=x^{-\alpha}e^{\beta x}$ is asymptotic to $x^{-\alpha}$. Its integral from zero diverges exactly when $\alpha\geq1$; in that case $s(0+)=-\infty$, and Feller’s test makes zero inaccessible.

Check accessibility when $\alpha<1$.
Then $s(0+)$ is finite and $s(x)-s(0+)$ is asymptotic to $x^{1-\alpha}/(1-\alpha)$. Since $m(x)$ is asymptotic to $(2/\sigma^2)x^{\alpha-1}$, their product is asymptotic to the finite constant $2/[\sigma^2(1-\alpha)]$. The boundary integral is therefore finite, so zero is accessible.

Translate the exponent.
$\alpha=2\kappa\theta/\sigma^2$. Thus $\alpha\geq1$ is exactly $2\kappa\theta\geq\sigma^2$; $\alpha<1$ is exactly the reverse inequality. This proves both directions of the stated classification. $\square$

The same threshold can be written $\kappa\theta\geq\sigma^2/2$. Equality belongs to the inaccessible case: from a positive initial state the process does not hit zero. This does not mean it stays a fixed positive distance away from zero; it may visit arbitrarily small positive values.

Worked Parameter Checks

Worked Example: A Concentration Process With Inaccessible Zero

Suppose $X_t$ is measured in mg/L, with $\kappa=0.8$ day$^{-1}$, $\theta=10$ mg/L, and $\sigma=3\sqrt{\text{mg/L}}/\sqrt{\text{day}}$. Check the condition.

First, $2\kappa\theta=2(0.8)(10)=16$ (mg/L)/day. Also, $\sigma^2=3^2=9$ (mg/L)/day. Hence $16\geq9$, and zero is inaccessible from $X_0>0$. Equivalently, $\alpha=16/9$, which exceeds one.

The units are consistent: $\sqrt{X_t}$ has units $\sqrt{\text{mg/L}}$, so $\sigma\sqrt{X_t}$ has units mg/L/$\sqrt{\text{day}}$; multiplication by $dW_t$, with units $\sqrt{\text{day}}$, gives concentration units.

Worked Example: A Process That Can Reach Zero

Let $\kappa=0.5$ day$^{-1}$, $\theta=2$ mg/L, and $\sigma=2\sqrt{\text{mg/L}}/\sqrt{\text{day}}$. Then $$ 2\kappa\theta=2(0.5)(2)=2\ \text{(mg/L)/day},\qquad \sigma^2=4\ \text{(mg/L)/day}. $$

Because $2<4$, the Feller condition fails. Here $\alpha=2/4=0.5<1$, and zero is accessible from a positive starting value. This does not mean that every path hits zero, or that it hits zero by any particular observation time; it means the continuous-time model permits boundary hitting with positive probability.

Worked Example: Translating the Condition Into a Noise Limit

Suppose $\kappa=0.3$ day$^{-1}$ and $\theta=5$ mg/L. The inequality requires $$ \sigma^2\leq2\kappa\theta=2(0.3)(5)=3\ \text{(mg/L)/day}. $$ Thus the largest diffusion scale satisfying the condition is $\sqrt{3}\sqrt{\text{mg/L}}/\sqrt{\text{day}}$. If instead $\sigma=2\sqrt{\text{mg/L}}/\sqrt{\text{day}}$, then $\sigma^2=4$ in the same units, so zero is accessible; $\alpha=3/4$. A value $\sigma=1.5\sqrt{\text{mg/L}}/\sqrt{\text{day}}$ gives $\sigma^2=2.25$, so the condition holds and zero is inaccessible.

This calculation can be useful when the mean-reversion parameters are scientifically motivated and the noise scale is estimated. It identifies the value of $\sigma$ beyond which the fitted continuous model no longer rules out zero hitting.

Interpreting the Condition in Pharmacokinetics

The CIR structure is plausible when a positive state has mean-reverting dynamics and its local variance increases in proportion to its current level. Near zero, the relative roles of the two coefficients matter: the drift at zero is $\kappa\theta$, while the squared diffusion scale is $\sigma^2$. A larger $\kappa\theta$ or smaller $\sigma^2$ favors inaccessibility. A larger noise scale relative to the mean-reverting drift allows the process to reach zero.

This is a model property, not a universal pharmacokinetic law. A concentration that is exactly zero between doses, a process with deterministic elimination to zero, or a state with dynamics not well described by mean reversion and variance proportional to concentration may not be represented plausibly by this CIR model. In particular, if $\theta=0$, then $2\kappa\theta=0<\sigma^2$; zero is accessible, and it is absorbing for the nonnegative CIR solution. That boundary behavior may not represent a system in which drug input reliably restarts positive concentration.

The CIR model is one alternative to additive-noise diffusions, which can cross below zero, and to positive multiplicative-noise models, which have different state-dependent variance and boundary behavior. These alternatives encode different assumptions; choosing the CIR form does not follow merely from wanting simulated concentrations to be positive. Nor does the Feller condition validate the whole pharmacokinetic model: it only classifies the zero boundary for this particular diffusion between any external interventions.

Diagnostics, Limitations, and a Common Mistake

Check parameter units. Use $\sigma$ in square-root-of-concentration per square-root-time units. Comparing $2\kappa\theta$ with $\sigma^2$ is dimensionally valid because both have concentration-per-time units.
Check the initial state and boundary claim. The theorem concerns hitting zero from $X_0>0$. It does not assert that a process initialized at zero is positive at its initial instant.
Do not infer the continuous boundary from a finite simulation. A path that stays positive on a chosen grid does not establish inaccessibility. Conversely, discretization can produce behavior that the continuous model does not have.
Do not overinterpret inaccessibility. It rules out hitting zero, not approaching it closely, and it does not guarantee that a concentration stays above a clinically meaningful threshold.

A common mistake is to argue that zero must be inaccessible because $\sigma\sqrt{x}$ tends to zero as $x$ tends to zero. That reasoning considers the diffusion coefficient but ignores the rate at which its noise acts over time relative to the drift and the boundary’s scale behavior. The correction is to calculate $\alpha=2\kappa\theta/\sigma^2$: zero is inaccessible exactly when $\alpha\geq1$, not simply because the diffusion coefficient vanishes at zero.

In practice, inspect the fitted ratio $2\kappa\theta/\sigma^2$, verify units and parameter constraints, and ask whether the implied boundary behavior matches the biology and observation process. When estimates lie near the threshold, uncertainty in the parameters matters: a point estimate on one side does not establish that the true process has that classification with certainty.

Check Your Understanding

Answer each question, then compare with the solutions below.

For $\kappa=0.4$ day$^{-1}$, $\theta=3$ mg/L, and $\sigma=1.5\sqrt{\text{mg/L}}/\sqrt{\text{day}}$, is zero inaccessible from a positive state?
If $2\kappa\theta=\sigma^2$, is zero accessible or inaccessible from $X_0>0$?
Why does the fact that $\sigma\sqrt{x}$ tends to zero as $x$ tends to zero fail, by itself, to prove zero inaccessibility?
What boundary classification applies when $\theta=0$, $\kappa>0$, and $\sigma>0$?

Solutions. (1) $2\kappa\theta=2(0.4)(3)=2.4$ (mg/L)/day, while $\sigma^2=1.5^2=2.25$ (mg/L)/day. Since $2.4\geq2.25$, zero is inaccessible. (2) Equality satisfies the Feller condition, so zero is inaccessible from a positive initial state. (3) Boundary accessibility depends on the combined scale and speed behavior, equivalently on the ratio $2\kappa\theta/\sigma^2$; a vanishing diffusion coefficient alone does not decide it. (4) The inequality fails because $2\kappa\theta=0<\sigma^2$, so zero is accessible; for the nonnegative CIR solution it is absorbing.