Square-Root Diffusion Coefficients and Model Forms

What Makes a Diffusion a Square-Root Model?

In a diffusion model, the coefficient multiplying Brownian noise determines the instantaneous scale of random fluctuations. A square-root diffusion is one in which that coefficient is proportional to the square root of the current state. This form is useful when the state is nonnegative and its instantaneous variance is proportional to its level: fluctuations become smaller as the state approaches zero.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a standard Brownian motion $B_t$. For an Ito integral $\int_0^t H_s\,dB_s$, the integrand $H_s$ is understood to be progressively measurable and in $\mathcal{L}^2$ on each finite interval under discussion; that is, $\mathbb{E}\int_0^T H_s^2\,ds<\infty$. In the models below, this condition is met on $[0,T]$ when $\mathbb{E}\int_0^T X_s\,ds<\infty$. More generally, the equations may be considered locally, up to stopping times on which the corresponding integrability condition holds.

Definition: A one-dimensional square-root diffusion on the state space $[0,\infty)$ has an equation of the form $$ dX_t=\mu(t,X_t)\,dt+c(t)\sqrt{X_t}\,dB_t, $$ where $c(t)$ is a specified nonnegative coefficient and $\mu$ is the drift. The diffusion coefficient is $c(t)\sqrt{x}$; its square, the instantaneous variance coefficient, is $c(t)^2x$. The state-space declaration specifies the states under consideration, not by itself whether the process can reach or leave the boundary $0$.

For a time-homogeneous model with constant $c$, the local variance is proportional to the state. The coefficient itself is not proportional to $x$: it is proportional to $\sqrt{x}$. This distinction matters in both calculations and interpretation. The drift controls the conditional first-order change, while the diffusion coefficient controls the conditional second-order fluctuation.

The CIR and Squared Bessel Model Forms

The Cox–Ingersoll–Ross (CIR) model has mean-reverting affine drift and a square-root diffusion coefficient. Its conventional parameterization uses $\kappa>0$, $\theta\ge0$, and $\sigma\ge0$, with initial state $X_0=x_0\in[0,\infty)$:

dX_t=\kappa(\theta-X_t)\,dt+\sigma\sqrt{X_t}\,dB_t,\qquad X_t\in[0,\infty).

Here $\kappa$ is the rate in the mean-reverting drift, $\theta$ is its level, and $\sigma$ scales the diffusion. The drift can also be written as $\kappa\theta-\kappa X_t$, making its affine form explicit. For a state $x\ge0$, the diffusion coefficient is $\sigma\sqrt{x}$, and the local variance coefficient is $\sigma^2x$.

The squared Bessel process of dimension $\delta\ge0$ is conventionally written, for an initial state $R_0=r_0\in[0,\infty)$, as

dR_t=\delta\,dt+2\sqrt{R_t}\,dB_t,\qquad R_t\in[0,\infty).

The constant drift is $\delta$, and the diffusion coefficient is $2\sqrt{r}$, so the local variance coefficient at state $r$ is $4r$. These equations specify the standard model forms and their state spaces. A declaration of $[0,\infty)$ does not assert strict positivity, nor does it settle whether zero is attainable; boundary behavior requires additional analysis.

Recognizing the CIR Form

Proposition: Suppose $a,b,c$ are constants with $a>0$, $b>0$, and $c\ge0$. The equation $$ dX_t=(a-bX_t)\,dt+c\sqrt{X_t}\,dB_t,\qquad X_t\in[0,\infty), $$ is a CIR equation with parameters $\kappa=b$, $\theta=a/b$, and $\sigma=c$. These parameters are the unique ones in the conventional CIR form with $\kappa>0$ that give the same drift and diffusion coefficients.

Proof. Set $\kappa=b$, $\theta=a/b$, and $\sigma=c$. The assumptions $a>0$ and $b>0$ give $\kappa>0$ and $\theta>0$, while $c\ge0$ gives $\sigma\ge0$. Substitution into the CIR drift gives $\kappa(\theta-x)=b(a/b-x)=a-bx$, and its diffusion coefficient is $c\sqrt{x}$, as required. Conversely, if the coefficients agree with a CIR equation for all $x\ge0$, equality of the affine drift coefficients gives $\kappa\theta=a$ and $\kappa=b$, so $\theta=a/b$; equality of the diffusion coefficients gives $\sigma=c$. Thus these parameters are unique. $\square$

The condition $b>0$ is needed for the stated mean-reverting CIR parameterization, and the ratio $a/b$ is defined here because $b>0$. This coefficient matching does not prove a solution exists or establish its boundary behavior; it identifies the equation's model form.

Worked Example: Identify a CIR Model

Write the model $dX_t=(6-2X_t)\,dt+3\sqrt{X_t}\,dB_t$ in CIR notation and give its state-dependent diffusion and variance coefficients.

Match the drift $6-2x$ with $a-bx$: $a=6$ and $b=2$. Match the diffusion coefficient $3\sqrt{x}$, which gives $c=3$. Therefore $\kappa=b=2$, $\theta=a/b=6/2=3$, and $\sigma=c=3$. The CIR form is

dX_t=2(3-X_t)\,dt+3\sqrt{X_t}\,dB_t,\qquad X_t\in[0,\infty).

At a state $x\ge0$, the diffusion coefficient is $3\sqrt{x}$ and its square is $9x$. Check: expanding the CIR drift gives $2(3-x)=6-2x$, and its diffusion coefficient $3\sqrt{x}$ matches the original equation.

Generator and Quadratic Variation

The generator records the drift of a smooth function of the state, including the correction contributed by Brownian fluctuations. For the affine-drift square-root equation $dX_t=(a-bX_t)\,dt+c\sqrt{X_t}\,dB_t$, define the generator on twice continuously differentiable functions $f$ by

Theorem: At $x\ge0$, the generator of the affine-drift square-root diffusion is $$ \mathcal{L}f(x)=(a-bx)f'(x)+\frac12c^2x f''(x). $$ For any solution for which the Ito integrals are defined, Ito's formula gives $$ df(X_t)=\mathcal{L}f(X_t)\,dt+c\sqrt{X_t}f'(X_t)\,dB_t. $$

Proof. Assume $f\in C^2$ on the state range of $X$, and apply Ito's formula to $f(X_t)$. For $f$ with no explicit time dependence, the first derivative is $f'(X_t)$, the second derivative is $f''(X_t)$, and the quadratic-variation correction is $\tfrac12 f''(X_t)\,d[X]_t$. The diffusion part of $X$ is the Ito integral with integrand $c\sqrt{X_s}$, so its quadratic variation is $d[X]_t=c^2X_t\,dt$. Substituting the drift and this quadratic variation into Ito's formula gives $$ df(X_t)=f'(X_t)(a-bX_t)\,dt+c\sqrt{X_t}f'(X_t)\,dB_t+\frac12 f''(X_t)c^2X_t\,dt. $$ Collecting the two $dt$ terms gives the stated generator. The correction has a plus sign and includes the factor $1/2$; it comes from $dB_t\,dB_t=dt$, while $dB_t\,dt=0$ and $dt\,dt=0$. $\square$

The same diffusion coefficient identifies the local quadratic variation: $d[X]_t=c^2X_t\,dt$. Thus, over an interval, the accumulated quadratic variation is the time integral of the state-dependent variance coefficient. It is not the ordinary variance of $X_t$, and the drift term $a-bX_t$ does not contribute to it.

Worked Example: Generator of a CIR Model on a Quadratic Function

For the CIR model $dX_t=2(3-X_t)\,dt+3\sqrt{X_t}\,dB_t$, compute $\mathcal{L}f(x)$ for $f(x)=x^2$, then state the drift and diffusion terms in $d(X_t^2)$.

Here $a=6$, $b=2$, and $c=3$. For $f(x)=x^2$, the derivatives are $f'(x)=2x$ and $f''(x)=2$. Substitution into the generator gives

\mathcal{L}f(x)=(6-2x)(2x)+\frac12(3^2)x(2)=12x-4x^2+9x=21x-4x^2.

The Ito diffusion term is $c\sqrt{x}f'(x)=3\sqrt{x}(2x)=6x\sqrt{x}$. Therefore

d(X_t^2)=(21X_t-4X_t^2)\,dt+6X_t\sqrt{X_t}\,dB_t.

Check: applying Ito's formula directly gives $2X_t\,dX_t+(dX_t)^2$. The drift contribution from $2X_t\,dX_t$ is $2X_t(6-2X_t)\,dt= (12X_t-4X_t^2)\,dt$, and the correction is $(3\sqrt{X_t})^2dt=9X_t\,dt$. Their sum is $(21X_t-4X_t^2)\,dt$, as above.

Worked Example: Generator of a Squared Bessel Process

For $dR_t=5\,dt+2\sqrt{R_t}\,dB_t$, compute the generator on $f(r)=r^3$, and determine the local quadratic variation coefficient.

Here the drift is $5$, so $a=5$ and $b=0$; the diffusion scale is $c=2$. The derivatives are $f'(r)=3r^2$ and $f''(r)=6r$. Hence

\mathcal{L}f(r)=5(3r^2)+\frac12(2^2)r(6r)=15r^2+12r^2=27r^2.

The Ito diffusion term is $2\sqrt{r}(3r^2)=6r^{5/2}$, so $d(R_t^3)=27R_t^2\,dt+6R_t^{5/2}\,dB_t$. The local quadratic variation is $d[R]_t=(2\sqrt{R_t})^2dt=4R_t\,dt$. Check: the correction in Ito's formula is $\tfrac12(6R_t)(4R_t\,dt)=12R_t^2\,dt$; adding the drift contribution $15R_t^2\,dt$ gives $27R_t^2\,dt$.

Model Conventions and a Common Pitfall

The CIR and squared Bessel equations both have square-root diffusion coefficients, but their drifts are different. CIR has a mean-reverting affine drift $\kappa(\theta-X_t)$; squared Bessel has constant drift $\delta$. The squared Bessel equation is not obtained merely by replacing the CIR parameters with a dimension parameter: their model forms should be read from their respective coefficients.

Common mistake: treating the diffusion coefficient as the variance coefficient. For the CIR equation with $\sigma=3$, writing the variance coefficient as $3\sqrt{x}$ is incorrect: that is the diffusion coefficient. Squaring it gives the correct local variance coefficient $(3\sqrt{x})^2=9x$, and therefore $d[X]_t=9X_t\,dt$. Conversely, using $9\sqrt{x}$ as the diffusion coefficient would produce variance coefficient $81x$, not $9x$.

Another important distinction is between the chosen state space and a boundary theorem. Both model forms are conventionally specified on $[0,\infty)$, where their square roots are real. This does not alone prove that a solution remains nonnegative, that it can reach zero, or that it stays strictly positive. Those are separate questions about solutions and boundary behavior, not consequences of matching coefficients or calculating the generator.

Model	Drift	Diffusion coefficient	State space
CIR	$\kappa(\theta-x)$	$\sigma\sqrt{x}$	$[0,\infty)$
Squared Bessel	$\delta$	$2\sqrt{x}$	$[0,\infty)$
Affine-drift square-root form	$a-bx$	$c\sqrt{x}$	$[0,\infty)$

Check Your Understanding

Use coefficient matching and the generator formula to answer each question. State any parameter restrictions needed for your conclusion.

Write $dX_t=(8-4X_t)\,dt+2\sqrt{X_t}\,dB_t$ in CIR notation.
For $dR_t=3\,dt+2\sqrt{R_t}\,dB_t$, calculate the local quadratic variation coefficient.
For a square-root diffusion with drift $a-bx$ and diffusion coefficient $c\sqrt{x}$, calculate the generator applied to $f(x)=x$.
Explain why specifying the state space $[0,\infty)$ does not establish that a process is strictly positive.
For $dX_t=(a-bX_t)\,dt+c\sqrt{X_t}\,dB_t$, identify the Ito correction when applying Ito's formula to $f(x)=x^2$.

Model	Drift	Diffusion coefficient	State space
CIR	\(\kappa(\theta-x)\)	\(\sigma\sqrt{x}\)	\([0,\infty)\)
Squared Bessel	\(\delta\)	\(2\sqrt{x}\)	\([0,\infty)\)
Affine-drift square-root form	\(a-bx\)	\(c\sqrt{x}\)	\([0,\infty)\)