Let $$S_t^2=S_0^2+2\int_{0}^{t}\sigma_uS_u^2\,\mathrm dW_u+\int_{0}^{t}\sigma_u^2S_u^2\,\mathrm du,$$ where $\sigma$ is bounded and predictable, $t\geq 0, S_0\in\mathbb R$.
Let $\tau_n:=\inf\{t\geq 0:|S_t|=n\}$.
Why do we have $$\mathbb E[S_{t\land\tau_n}^2]=S_0^2+\int_0^t\mathbb E[\sigma_u^2S_{u\land\tau_n}^2]\,\mathrm du$$ i.e. why does the integral w.r.t $\mathrm dW_u$ vanish in expectation?
The process $$M_{t \land \tau_n} = \int_0^{t \land \tau_n}\sigma_u S_u^2 dW_u = \int_0^{t}\sigma_u S_u^2 1_{\{u \leq \tau_n\}}dW_u $$ is a martingale. This follows because the integrand $\sigma_u S_u^2 1_{\{u \leq \tau_n\}}$ is bounded, by the definition of $\tau_n$ and since $\sigma_u$ is bounded. Hence $M_{t\land \tau_n}$ has constant mean, but since it starts at zero, that mean must be zero.