Suppose there is a second-order real-state stochastic process $X: \Omega \times T \rightarrow \mathbb{R}$ with $T= \mathbb{R}$ and probability space $(\Omega, \mathcal{F}, P)$. I was wondering if the following inequality holds for integrals over an interval $[a,b] \subset \mathbb{R}$:
$ \Vert \int_a^b X_t dt \Vert_2 \leq \int_a^b \Vert X_t \Vert_2 dt \ ?$
Note that the integral on LHS is a stochastic integral while the one on RHS is a deterministic one. $L_2$ norm is on $L_2$ space of squared-integrable random variables as measurable mappings from $(\Omega, \mathcal{F}, P)$ to $(\mathbb{R}, \mathcal{B})$.
Is this Jensen's inequality? I don't think it is, because although the $L_2$ norm is convex, it is not a mapping from $\mathbb{R}$ to $\mathbb{R}$, but from $L_2(\Omega, \mathcal{F})$ to $\mathbb{R}$, and the types of integrals on LHS and on RHS are not the same. So I was wondering if this inequality is true and why?
Can it be seen as a generalization of Jensen's inequality? If yes, is it only because of the similarity of their forms or the similarity of some deeper things?
Thanks and regards!