# 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)?

## 2 Answers

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 Nov 14, 2020 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! Nov 14, 2020 at 18:02