# Proof of Feller condition for CIR square root process. Any reference?

Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form: $$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$ with $$\left(B_t\right)_{t\geq0}$$ denoting a Brownian motion on the filtered probability space $$\left(\Omega,\mathcal{F},\mathcal{F}_n,\mathbb{P}\right)$$?

I found something here, but I cannot really understand it, hence I am searching for something alternative. Is there some alternative proof (e.g. from a book)?

It is covered very nicely in Iain Clark's Foreign Exchange Option Pricing, A Practitioner’s Guide (pages 98-104). The book also contains references to the relevant literature including Feller's original paper.

I believe your SDE has an unintended error. It should be:

$$dr_t = a \cdot (b - r_t) \cdot dt + \sigma \cdot \sqrt{r_t} \cdot dB_t.$$

On the other hand, the Feller condition is discussed and explained in Section 10.2.1.2 (pg. 432) of the Andersen and Piterbarg book: Interest Rate Modeling.

Hope it helps!

• A silly error of mine, of course, sorry. I have just edited the question. Anyway, I cannot find the proof in your reference, but just a statement of the Feller's condition – Strictly_increasing Nov 14 '20 at 17:11
• Yes, you are right. It doesn't have a full proof there, but the comments are really helpful. It also references Proposition 8.3.1, in Section 8.3 (pg. 319). There, I think it references the paper "Moment explosions in stochastic volatility models" from Andersen and Piterbarg or the book "Continuous Martingales and Brownian Motion" from Revuz and Yor. Thank you! – rvignolo Nov 14 '20 at 18:02