Worked Boundary Cases for CIR, Bessel, and CEV Models

Reading Boundary Cases Without Confusing Reachability and Absorption

For a diffusion on the nonnegative half-line, three different questions matter: can the process reach zero in finite time, what happens if it reaches zero, and is zero inaccessible from a positive starting point? These questions are related, but they are not interchangeable. In particular, a process can reach zero and then leave it, or reach zero and remain there forever. The calculations below use parameter criteria already established in the earlier material on CIR boundary behavior, Bessel processes at the origin, and the CEV-to-Bessel transformation.

All SDEs are considered on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) carrying Brownian motion \(B_t\). An SDE is understood in its integral form, with progressively measurable Ito integrands. On finite time intervals where the integrands are square-integrable, they belong to \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). At a boundary or where a coefficient is only locally controlled, the equation is interpreted up to suitable stopping times; the stopped integrands are required to be in \(\mathcal{L}^2\). The drift and diffusion coefficients are distinct: the drift multiplies \(dt\), and the diffusion coefficient multiplies \(dB_t\).

CIR: Apply the Feller Ratio Carefully

Consider the CIR equation \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\sqrt{X_t}\,dB_t\), with \(X_0=x_0>0\), \(\kappa>0\), \(\theta>0\), and \(\sigma>0\). The relevant ratio is \(2\kappa\theta/\sigma^2\). The established CIR boundary classification says that zero is inaccessible when this ratio is at least \(1\), including equality, and attainable when it is less than \(1\). When it is attainable and \(\theta>0\), the standard CIR process is not absorbed at zero: its positive drift there, \(\kappa\theta\), points into the state space.

Proof. The CIR-to-squared-Bessel time-change uses dimension \(\delta=4\kappa\theta/\sigma^2\). Dividing this equality by \(2\) gives \(\delta/2=2\kappa\theta/\sigma^2\). Since division by the positive number \(2\) preserves order, \(\delta\geq2\) holds if and only if \(2\kappa\theta/\sigma^2\geq1\); likewise, \(\delta<2\) holds if and only if \(2\kappa\theta/\sigma^2<1\). The squared-Bessel origin classification therefore gives the stated equivalent boundary test. \(\square\)

Squared Bessel: Dimension Determines the Origin Behavior

The squared-Bessel process \(Z_t\) of dimension \(\delta\geq0\), started at \(z_0>0\), satisfies \(dZ_t=\delta\,dt+2\sqrt{Z_t}\,dB_t\). The standard origin classification is: zero is inaccessible for \(\delta\geq2\); for \(0<\delta<2\), it is attainable and instantaneously reflecting; for \(\delta=0\), it is attainable and absorbing. “Instantaneously reflecting” means that the process can hit zero but does not stay there for a positive interval as an absorbed process would.

CEV: Transform the Exponent, Then Inspect the Boundary Rule

Consider \(dS_t=\mu S_t\,dt+\sigma S_t^\beta\,dB_t\), with \(S_0=s_0>0\), \(\sigma>0\), and \(0<\beta<1\). Before zero is reached, put \(p=1-\beta>0\) and \(Y_t=S_t^p\). Ito’s formula gives a Bessel-type equation. The dimension parameter determines the local boundary classification, while the chosen boundary extension determines whether the process is absorbed or continues after a hit.

Proof. Apply Ito’s formula to \(f(s)=s^p\), with \(f'(s)=ps^{p-1}\) and \(f''(s)=p(p-1)s^{p-2}\). The correction term is \(\tfrac12 f''(S_t)\sigma^2S_t^{2\beta}\,dt\). Since \(p=1-\beta\), this gives \(dY_t=p\mu Y_t\,dt-\frac{p\beta\sigma^2}{2Y_t}\,dt+p\sigma\,dB_t\). Dividing by \(p\sigma\) and using \(Y_t=p\sigma R_t\) gives the displayed equation for \(R_t\). A Bessel drift has singular term \((\delta-1)/(2R_t)\). Equating it to \(-\beta/(2(1-\beta)R_t)\) yields \(\delta-1=-\beta/(1-\beta)\), hence \(\delta=(1-2\beta)/(1-\beta)\). Because \(1-\beta>0\), \(\delta<2\) is equivalent to \(1-2\beta<2(1-\beta)=2-2\beta\), which reduces to \(1<2\), always true. The sign of \(\delta\) is the sign of \(1-2\beta\), proving the remaining claims. \(\square\)

The linear drift \((1-\beta)\mu R_t\) does not change this local classification at zero. For \(0<\beta<\tfrac12\), the positive dimension lies below \(2\), the reflecting Bessel-type continuation reaches and leaves zero. An absorbing CEV model can instead impose absorption there; that is a boundary choice, not a conclusion from attainability. For \(\tfrac12\leq\beta<1\), the transformed dimension is nonpositive, and the usual nonnegative CEV boundary behavior is absorption. At \(\beta=1\), the equation is geometric Brownian motion, whose positive solution does not hit zero in finite time.

Common Mistake: Dropping the Factor of Two

A frequent CIR error is to test \(\kappa\theta/\sigma^2\) against \(1\), omitting the factor \(2\). For \(\kappa=1\), \(\theta=\tfrac34\), and \(\sigma=1\), the incorrect ratio is \(1\cdot\tfrac34/1^2=\tfrac34\), which might be read as evidence that zero is attainable. The correct ratio is \(2\cdot1\cdot\tfrac34/1^2=\tfrac32\), so zero is inaccessible. The dimension check agrees: \(\delta=4\cdot1\cdot\tfrac34/1^2=3\), and \(\delta\geq2\).

Always state both parts of a boundary conclusion: whether zero can be reached, and what the specified model does after reaching it. In particular, an absorbing convention is not implied just by an accessibility calculation. The distinction is important when comparing models, selecting a boundary extension, or interpreting a simulated path that approaches zero.