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Feller Boundary Condition for the CIR Square-Root Process

Article Quant Q&A · Author: Strictly_increasing
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Summary

The document asks for references explaining the Feller condition for the Cox–Ingersoll–Ross square-root diffusion, a model often used for interest rates. It points readers to treatments in Iain Clark’s Foreign Exchange Option Pricing and Andersen and Piterbarg’s Interest Rate Modeling, as well as references to Feller’s original work.

One response also flags a likely notation error in the question’s stochastic differential equation and gives the intended mean-reverting drift form. The post itself does not provide a proof or explain the condition’s mathematical implications; its value is directing readers to book treatments. The references are suggestions rather than a worked derivation, so readers seeking an accessible proof will need to consult those sources.

Key ideas

  • The question concerns the Feller condition for a CIR square-root diffusion.
  • The replies recommend textbooks on foreign exchange option pricing and interest-rate modeling.
  • One answer corrects the equation’s likely intended mean-reverting drift term.
  • The post supplies references but no proof or detailed explanation.

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Fixed incomeStatistics
Full text
# Proof of Feller condition for CIR square root process. Any reference?


# 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)?

## Answer by Magic is in the chain (score 6)

https://quant.stackexchange.com/a/59372

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.

## Answer by rvignolo (score 3)

https://quant.stackexchange.com/a/59369

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!

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