I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$ \mathrm{d}\mathrm{e}^{-r(T-t)}V(S_t,t)= \mathrm{e}^{-r(T-t)}\sigma S_t\frac{\partial V}{\partial S}(S_t,t)\mathrm{d}W_t $$
The Ito integral is a martingale if
$$ \mathbb{E}\left[\int_0^T\left(S_t\frac{\partial V}{\partial S}(S_t,t)\right)^2\right]<\infty $$
Unfortunately, I am not able to show this as I cannot apply Jensen, Hölder or Cauchy-Schwartz to eliminate the square. How do I get around this issue. A related question is whether the delta for an arbitrary option is bounded in the BS-model.