ARCH Model - Expectation of Absolute Value
I don't believe $w&gt;0$ is necessary or sufficient to get a positive standard deviation everywhere. I think you need $\sigma_0 \geq 0$ and $w\geq 0$ with one of the two inequalities strict.

If $\gamma  &gt; 1$ then the series $\sigma$ will grow explosively if $w + \gamma \cdot \sigma_{t-1}  &gt;0$ or behave explosively harmonic if $w + \gamma \cdot \sigma_{t-1} &lt;0 $. Neither of those seems particularly desirable most for economic situations.