# 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:

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

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!

• See this paper for discussion. Nov 17 '17 at 17:12
• @Gordon Thank you so much, very clear proof (but not so trivial as everybody says). Nov 18 '17 at 17:54