Consider the stochastic integral \begin{equation} \int_0^t f(s) \mathrm{d}W_s \end{equation} for a not necessarily deterministic function $ f$ . Can I bound this random variable in second order stochastic dominance $ \leq_\text{(2)}$ by an appropriately scaled normal distributed random variable?

Here random variables $ X,Y$ satisfy \begin{equation} X \leq_\text{(2)} Y \end{equation} if and only if for all $ q \in \mathbb{R}$ \begin{equation} \int_{-\infty}^q F_X(x) \mathrm{d}x \geq \int_{-\infty}^q F_Y(y) \mathrm{d}y \end{equation} holds. For simplicity one might assume that $ f$ is bounded.