The CIR Feller Condition and Zero-Boundary Inaccessibility

Summary

The document states the Feller condition for the Cox–Ingersoll–Ross short-rate process: when twice the mean-reversion speed times the long-run mean is at least the squared volatility, the process remains nonnegative and does not reach zero from a positive starting value. The answer sketches a boundary-hitting argument using a scale function and stopping times for reaching a lower level or an upper threshold.

Under the condition, the scale function diverges as the lower level approaches zero, implying that the probability of hitting zero before an upper level is zero; the answer then extends this to zero being unreachable in finite time. The material supplies a proof outline, not a bibliographic citation, despite the original question asking for a source. Its notation and boundary conclusion presume positive model parameters and an initially positive rate; it does not discuss other parameter regimes or whether nonnegativity alone holds without the condition.

Key ideas

• The CIR short-rate process has square-root volatility and mean reversion toward a long-run level.
• The stated Feller condition is twice the mean-reversion speed times the long-run mean being at least the squared volatility.
• A scale-function argument relates the condition to the probability of hitting a lower boundary before an upper one.
• When the condition holds, a process starting above zero does not hit zero in finite time.
• The answer outlines reasoning but does not provide the requested citation.

Tags

# Feller Condition (Cox-Ingersoll-Ross) source


# Feller Condition (Cox-Ingersoll-Ross) source












For the Cox-Ingersoll-Ross model $$\text{d}r_t = a(b-r_t)\text{d}t+\sigma\sqrt{r_t}\text{d}W_t$$ the condition (referred to as "Feller condition") $$2ab\geq\sigma^2$$ ensures that the solution is bounded below by zero. I see it being used all the time but I can't find a good source for citation... Can anybody help me out here?

## Answer by Greyearl (score 3)

https://quant.stackexchange.com/a/70897

For $x,K > 0 $, denote by $(X_t^x)_{t\geq 0}$ the unique strong solution of the CIR sde starting from $x$ at time 0.

Define the following stopping time :

$$\tau_{K}^x := \inf\left\{t\geq 0 : \quad X_t^x = K \right\} $$

Let us define furthermore the function $\psi$ defined on $\mathbb{R}_+^*$ by : $$ \forall \ x > 0 : \quad \psi(x) := \int_{1}^{x} y^{-\frac{2ab}{\sigma^2}}\exp(\frac{2ay}{\sigma^2})dy$$ For $0< \varepsilon < x < K$, define the following stopping time : $\tau_{\varepsilon, K}^x = \min(\tau_K^x, \tau_{\varepsilon}^x)$. Then, one can easily prove that $\tau_{\varepsilon, K}^x$ is a.s finite and for all $x \in (\varepsilon, K)$ we have : $$ \psi(x) = \psi(\varepsilon)\mathbb{P}\left(\tau_{\varepsilon}^x < \tau_K^x\right) + \psi(K) \mathbb{P}\left(\tau_{\varepsilon}^x > \tau_K^x\right)$$ Now, suppose that the feller condition is satisfied. Then, noticing that $\displaystyle \lim_{x\to 0^+}\psi(x) = - \infty$, one can prove that : $$\forall K> 0 : \quad \mathbb{P}\left(\tau_{0}^x < \tau_K^x\right) = 0$$ And so that : $$ \mathbb{P}\left(\tau_{0}^x < \infty\right)=0$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.