I'm trying to derive the estimate $$ E\left[\left|\int_{0}^{t}h_r\,dB_r\right|^4\right] \leq 3C^4t^2,$$ where $h_r$ is continuous, adapted (to the natural Brownian filtration up to time $t$) and bounded (with constant $C$).
I almost got this estimate, except that I have no idea how to infer that the expectation $$ E\left[\int_{0}^{t}X_r^3 h_r\, dB_r\right]$$ is zero (if it were always negative, it would suffice as well, but I highly doubt this is the case here). To get this far I applied the Ito's formula to $dX_t=h_tdB_t$ and $f(x)=x^4$.
I think you can use the Wiener integral to solve that problem ?
$\int_{0}^{t}h_rdB_r$~ $ \mathscr{N}(0,\int_{0}^{t}h_r^2dr)$ as h is continuous adapted to the filtration (Wiener integral)
Then using the 4th moment of a central normal distribution,
$E[\int_{0}^{t}h_rdB_r] = 3*(\int_{0}^{t}h_r^2dr)^2 < 3C^4t^2$ using the fact that h is bounded.