Why is the Schöbel-Zhu model affine?

In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=kappa(theta-v_t)dt+sigma dW_t$.

The characteristic function of the stock process can be found by arguing that the model is affine if we expand the set of variables to include $v^2_t$.

The argument is that applying Ito’s lemma we find $dv^2_t=[2kappa(theta v_t – v^2_t)+sigma^2]dt+2sigma v_tdW_t$, which is rewritten as $dv^2_t=[2kappa(theta v_t – v^2_t)+sigma^2]dt+2sigma sqrt{v^2_t}dW_t$.

The second equation for $dv^2_t$ is indeed affine in $v_t$ and $v^2_t$. However it seems to me that it is really equal to $dv^2_t=[2kappa(theta v_t – v^2_t)+sigma^2]dt+2sigma |v_t|dW_t$, which is not equal to the first equation since $v_t$ can become negative.

Therefore, why is the formula derived by Schöbel and Zhu correct?