The CIR Positivity Condition and Its Relation to a Squared OU Process

Summary

The document asks about positivity of the Cox–Ingersoll–Ross square-root diffusion and the parameter restriction often used to ensure the process stays strictly positive. It introduces a construction in which an Ornstein–Uhlenbeck process is squared, producing a related square-root diffusion, and contrasts that intuition with the usual mean-reverting CIR notation.

The central question is whether the stated condition on the CIR parameters is sufficient for positivity and how to prove it. The excerpt contains no proof or answer. The squared-OU construction alone does not establish the general CIR condition: its parameters must be matched carefully, and strict positivity, nonnegativity, and boundary attainability are distinct claims. The document is therefore a theoretical question rather than a completed derivation.

Key ideas

• The CIR model is presented as a mean-reverting square-root diffusion.
• Squaring an Ornstein–Uhlenbeck process yields a related square-root process.
• The author asks about a parameter condition for strict positivity but provides no proof.
• A squared-OU representation does not by itself prove the general boundary condition for CIR.

Tags

# Proof positiveness condition CIR dynamic


# Proof positiveness condition CIR dynamic












Ciao All. I'm studying the CIR model and this question came out.

Usually the Ornstein-Uhnlenbeck dynamic is used to build the CIR model: let $$ dX_t = aX_t + \sigma dW_t $$ where $a \in \mathbb{R}$ and $\sigma >0$. Then if we call $Y_t = X_t^2$ we get: $$ dY_t = \left(\sigma^2 + 2aY_t \right) dt + 2 \sqrt{Y_t} \sigma dW_t $$ which is a CIR dynamic.

Starting from now we will use the usual notation:

\begin{equation} dY_t = a \left( b-Y_t \right)dt + \sigma \sqrt{Y_t} dW_t. \end{equation}

Of course $Y_t$ is positive a.e. since it's equal to $X_t^2$.

My question is about the condition $$ ab > 2 \sigma^2 $$

In fact according to many papers this is enough to make $Y_t$ positive. Can you give a proof of this fact?

Thank you! Ciao!

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