Does $\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$ holds?

Question

I was wondering if the inequality $$\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$$ holds for stochastic integral. In fact, I don't see such a property in any book, neither on Google, so I have some doubt. What do you think ?

Answer 1

Fails if $f(t,\omega)=1$ for all $t, \omega$.

Answer 2

I've always seen the stochastic integral as basically a probabilistic version of a Riemann-Stieltjes integral, where the main difference is that we have to use weak convergence in the definition.

Like in the Riemann-Stieltjes integral, we have to also enclose the $d\text{[blah]}$ part to get a true statement: $$\left|\int_0^T f(x) \, dg(x)\right| \leq \int_0^T \left|f(x)\, dg(x)\right|$$

Unfortunately, unlike in the Riemann-Stieltjes, $$\int_0^T \left|f(t,\omega) dW_t\right|$$ is only $0$ or $\infty,$ (when $W_t$ is Brownian, at least) depending on whether $f(\cdot, \omega) = 0\,\, a.e.$ or not, so it doesn't provide a useful result.