= Solution
For $S_0>0$, the <stochastic exponential> solution stays strictly positive. Put $U_t=\sqrt{S_t}$. The <Itô formula> gives
$$
\boxed{dU_t=\frac12U_t\sigma_t\,dW_t-\frac18U_t\sigma_t^2\,dt.}
$$
Thus $U$ is a nonnegative local <supermartingale>. To justify the true <supermartingale> property, stop where $U$ or its stochastic integral exceeds successive bounds. The stopped <Itô formula> gives $\mathbb E[U_{T\wedge\tau_m}\mid\mathcal F_t]\leq U_{t\wedge\tau_m}$ for $T\geq t$. The conditional <Fatou lemma> and $\tau_m\uparrow\infty$ yield
$$
\boxed{\mathbb E[U_T\mid\mathcal F_t]\leq U_t.}
$$
In particular $\mathbb EU_T\leq U_0<\infty$. The case $S_0=0$ is the identically zero process. This is the <square-root stock supermartingale>.
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