# Inequality used in proof of existence of SDE solutions?

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:

1. $$|b(t,x)| \leq C(1+|x|)$$
2. $$|b(t,x)-b(t,y)| \leq D|x-y|$$

for some constants $$C,D$$. But given neither appears in the inequality, I do not think they are used.

This inequality uses the (deterministic) Jensen's inequality, i.e.$$\left( \frac{1}{t} \int_0^t a(s, ω) \,\mathrm{d}s \right)^2 \leqslant \frac{1}{t} \int_0^t (a(s, ω))^2 \,\mathrm{d}s. \quad \forall t > 0,\ ω \in Ω$$ Multiplying by $$t^2$$ and taking expectations yields$$E\left( \left( \int_0^t a(s, ω) \,\mathrm{d}s \right)^2 \right) \leqslant t E\left( \int_0^t (a(s, ω))^2 \,\mathrm{d}s \right).$$