Positivity Claims Need Their Assumptions
In diffusion models, “nonnegative” and “strictly positive” are different conclusions. A process may stay in \([0,\infty)\) and still reach zero. A criterion proved for one model may not apply to another model, or even to a different initial state of the same model. For constant-elasticity-of-variance (CEV) models, the volatility exponent does not by itself specify what the process does at zero: the coefficient there and any boundary prescription matter.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Every stochastic integral below is an Itô integral. Its integrand is progressively measurable; when its second moment is finite on \([0,T]\), it belongs to \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Integrals that are only locally square-integrable are understood by stopping, and claims below are made on finite horizons where the stated solutions and integrals are defined.
For a continuous process, “positive at each fixed time” and “positive at every time on the interval” are distinct assertions: the former quantifies separately over each deterministic \(t\), while the latter concerns the whole path. Boundary theorems for the models below specify which assertion they establish.
The CIR Condition: What It Does and Does Not Say
The Cox–Ingersoll–Ross (CIR) equation is \(dX_t=\kappa(\theta-X_t)\,dt+\sigma\sqrt{X_t}\,dB_t\), with initial value \(X_0=x_0\). Here \(\kappa\) is the mean-reversion rate, \(\theta\) the long-run level, and \(\sigma\) the diffusion scale. The drift is \(\kappa(\theta-X_t)\); the diffusion coefficient is \(\sigma\sqrt{X_t}\).
The hypotheses are part of the result. In particular, the condition compares the CIR drift and diffusion parameters; it is not a universal test for positivity in other diffusions. Nor does the inequality alone guarantee strict positivity if the initial state is zero: the asserted pathwise conclusion on \([0,T]\) would already fail at time \(0\).
Worked Example: Nonnegative CIR with an Accessible Boundary
Take \(\kappa=1\), \(\theta=\tfrac14\), \(\sigma=1\), and \(x_0=1\). Check which CIR conclusion applies.
All three parameter assumptions hold: \(\kappa=1>0\), \(\theta=\tfrac14>0\), and \(\sigma=1>0\). The Feller comparison gives \(2\kappa\theta=2\cdot1\cdot\tfrac14=\tfrac12\), while \(\sigma^2=1^2=1\). Thus \(0<\tfrac12<1\). The standard classification says the process remains nonnegative, but zero is accessible and can be reached with positive probability.
Check: the strict-positivity condition \(2\kappa\theta\ge\sigma^2\) fails because \(\tfrac12\not\ge1\). That failure does not say the process becomes negative; the applicable classification still gives nonnegativity.
Worked Example: A Valid Feller Inequality with Zero Initial Value
Take \(\kappa=2\), \(\theta=1\), \(\sigma=1\), and \(x_0=0\). Does the inequality establish strict positivity on \([0,T]\)?
Compute \(2\kappa\theta=2\cdot2\cdot1=4\) and \(\sigma^2=1^2=1\), so \(4\ge1\). But \(X_0=0\), hence the event \(\{X_t>0\text{ for every }t\in[0,T]\}\) is empty: its condition already fails at \(t=0\). The positive-initial-state hypothesis of the strict-positivity result is absent.
Check: there is no contradiction with the CIR boundary classification. Under these parameters, the standard conclusion gives positivity at every \(t>0\) when starting from zero; it cannot give positivity on an interval that includes the zero initial value.
A frequent error is to treat the Feller inequality as a nonnegativity test. In the first example, that would incorrectly suggest that failure of the inequality permits negative values. The correction is to separate the two conclusions: for the stated CIR parameters, nonnegativity holds, while strict positivity fails as a general guarantee because zero is accessible.
CEV Models: Specify the Coefficient at Zero
A common CEV form, initially considered for \(S_t>0\), is \(dS_t=\mu S_t\,dt+\sigma S_t^\beta\,dB_t\), where \(\mu\) is the drift rate, \(\sigma>0\) is the diffusion scale, and \(\beta\) is the volatility exponent. To discuss a boundary at zero, the equation must be defined there as well. For \(\beta>0\), the diffusion coefficient extends continuously to zero with value zero, and the drift \(\mu S_t\) is also zero there. For \(\beta=0\), the natural constant-coefficient convention is \(S_t^0=1\), including at zero; the diffusion does not vanish there. For \(\beta<0\), \(S_t^\beta\) is singular at zero, so the displayed equation does not define a coefficient there.
Proof. Let \(Y_t\) solve the equation up to \(\tau\), and define \(S_t=Y_t\) for \(t\le\tau\) and \(S_t=0\) for \(t>\tau\). The drift coefficient at zero is \(\mu\cdot0=0\), and the diffusion coefficient is \(\sigma 0^\beta=0\), since \(\beta>0\). For any \(t\), the drift integral after \(\tau\) therefore adds zero. The stochastic integrand after \(\tau\) is also zero, so its Itô integral adds zero. Before \(\tau\), the equation holds because \(S=Y\). Combining the pre- and post-\(\tau\) integrals gives the CEV equation for \(S\). \(\square\)
Worked Example: An Absorbing CEV State at Zero
Take \(\beta=\tfrac12\), \(\mu=1\), \(\sigma=2\), and \(S_0=0\), with the coefficient defined on \([0,\infty)\). Verify the zero path.
Set \(S_t=0\) for every \(t\ge0\). At each time the drift is \(\mu S_t=1\cdot0=0\), and the diffusion coefficient is \(\sigma S_t^\beta=2\cdot0^{1/2}=0\). Thus, for each \(t\), the drift integral is \(\int_0^t0\,ds=0\) and the Itô integral is \(\int_0^t0\,dB_s=0\). Both integrands are progressively measurable and have \(\mathbb{E}\int_0^T H_s^2\,ds=0\).
Check: the right side of the integrated equation equals the initial value \(0\) plus \(0+0\), so it equals \(S_t=0\). This verifies a solution starting at zero; it does not decide whether a solution starting strictly above zero will hit the boundary.
Two CEV Exponents, Two Different Checks
Examples expose why “the exponent tells us whether zero is absorbing” is too crude. At \(\beta=1\), the model is geometric Brownian motion and stays strictly positive from a positive initial state. At \(\beta=0\), the constant diffusion coefficient can carry the process below zero. These statements concern the specified equations, not an exponent-only rule.
Worked Example: Geometric Brownian Motion Stays Positive
Let \(\beta=1\), \(\mu=0\), \(\sigma=1\), and \(S_0=1\). Verify the candidate \(S_t=\exp(B_t-\tfrac12t)\).
Apply Itô’s formula to \(f(t,x)=\exp(x-\tfrac12t)\), with \(S_t=f(t,B_t)\). Its derivatives are \(f_t=-\tfrac12f\), \(f_x=f\), and \(f_{xx}=f\). The full Itô calculation, including the second-derivative correction, is
This is the stated equation, and the exponential is strictly positive at every finite time. Its integrand is progressively measurable and is in \(\mathcal{L}^2\) on each finite horizon: since \(B_t\sim N(0,t)\), \(\mathbb{E}[S_t^2]=\mathbb{E}[\exp(2B_t-t)]=e^t\), so \(\mathbb{E}\int_0^T S_t^2\,dt=\int_0^T e^t\,dt=e^T-1<\infty\).
Check: the same normal moment calculation gives \(\mathbb{E}[S_t]=\mathbb{E}[\exp(B_t-\tfrac12t)]=1\), matching the zero-drift equation’s expected value. This independent check also confirms the exponential correction has the required sign.
Worked Example: Constant CEV Volatility Can Cross Zero
Let \(\beta=0\), \(\mu=0\), \(\sigma=1\), and \(S_0=1\), with \(S^0=1\) at zero. Solve the equation and test nonnegativity at \(t=1\).
The equation is \(dS_t=dB_t\), so its integrated solution is \(S_t=1+\int_0^t1\,dB_s=1+B_t\). The integrand \(H_s=1\) is progressively measurable and \(\mathbb{E}\int_0^T H_s^2\,ds=T<\infty\). Since \(B_1\sim N(0,1)\),
Thus the process is not nonnegative almost surely. At zero the diffusion coefficient is still \(1\), so this equation does not prescribe absorption there.
Check: substituting \(S_t=1+B_t\) into the integrated equation gives \(1+\int_0^t1\,dB_s=1+B_t\), as required. Restricting the state to \([0,\infty)\) would require an additional boundary rule; it would no longer be this unrestricted equation after a crossing.
A Reliable Boundary-Claim Checklist
Before using the words “positive,” “nonnegative,” “absorbing,” or “inaccessible,” check the model rather than relying on a remembered slogan. In particular, identify whether the claim is for each fixed time or for every time on a path, and whether the initial value satisfies the stated hypotheses.
Record the drift, diffusion coefficient, parameter restrictions, initial state, and state space. A condition from one model is not automatically valid for another.
Nonnegativity permits the value zero. Strict positivity excludes it. Check whether the claim includes time zero or concerns only positive times.
Evaluate the coefficients at zero when they are defined there. If they are singular or the state space excludes zero, the model needs a separate boundary specification before a conclusion can be drawn.
For CIR, check \(\kappa>0\), \(\theta>0\), \(\sigma>0\), \(x_0>0\), and \(2\kappa\theta\ge\sigma^2\) before invoking the stated strict-positivity result. For CEV, do not replace a boundary analysis with a statement about \(\beta\) alone.
The practical lesson is modest but important: positivity is a property of an equation together with its parameters, initial state, and boundary convention. A correct inequality applied outside its hypotheses, or a correct exponent interpreted without defining the model at zero, can still produce a false conclusion.
Check Your Understanding
For each question, state the assumptions you use and distinguish conclusions about nonnegativity from those about strict positivity.
- For a CIR process with \(\kappa=3\), \(\theta=\tfrac12\), and \(\sigma=1\), compute both sides of the Feller comparison. What can be concluded if \(x_0>0\)?
- Why does \(2\kappa\theta\ge\sigma^2\) not establish strict positivity on an interval including time zero when \(x_0=0\)?
- For the CEV equation with \(\beta>0\), explain why the coefficients vanish at zero and what that permits one to prescribe after a hit.
- For \(\beta=0\), \(\mu=0\), and \(\sigma=1\), what is the probability that the solution started at \(1\) is negative at time \(t>0\)?
- List the model information, beyond the CEV exponent, that should be checked before making a boundary claim.