In the proof of existence/uniqueness of SDE the following inequality is used:
$$E\left[ \left( \int_0^t a(s,\omega) ds \right)^2 \right] \leq t E\left[ \int_0^t a(s,\omega)^2 ds \right]$$
and I cannot really see how it is obtained. Here, $a$ is defined as $$a(s,\omega) = b(s,X_1(s)) - b(s,X_2(s)),$$ where $X_1$ and $X_2$ are stochastic processes, both of which satisfy the SDE $$dX_t = b dt + \sigma dBt,$$ and $b$ is assumed to satisfy:
- $|b(t,x)| \leq C(1+|x|)$
- $|b(t,x)-b(t,y)| \leq D|x-y|$
for some constants $C,D$. But given neither appears in the inequality, I do not think they are used.