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?
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3$\begingroup$ There is a comprehensive discussion in Chapter 6 of the book mathematical methods for financial markets. $\endgroup$– GordonMar 8, 2019 at 15:59
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1$\begingroup$ See also this paper. $\endgroup$– GordonMar 8, 2019 at 17:53
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3$\begingroup$ In the original CIR paper A Theory of the Term Structure of Interest Rates reference is made to Two Singular Diffusion Problems by Feller. Feller has all the details but it's not yet clear how to map the result there to the CIR paper. $\endgroup$– Bob Jansen ♦Mar 9, 2019 at 8:28
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$\begingroup$ Thanks very much! :) $\endgroup$– AlvoMar 11, 2019 at 12:54
1 Answer
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 : $$ \mathbb{P}\left(\tau_{0}^x < \infty\right)=0$$