Supremum of Stochastic processes

Question

Let $X(t)$ be a stochastic process. Let $A=E[\sup_{t \in T} X(t)]$ and $B= \sup_{t \in T} E[ X(t)]$. Can we establish any relation between A and B (Is there any? ) for e.g. $A>B \text{ or } A<B$. Prove the relation also.

Answer 1

The functional $F[X]=\max_{t \in T} X(t)$ is convex. So by Jensen's inequality, $E[F[X]] \geq F[E[X]]$. We can have strict inequality: for instance we do when $X$ is Brownian motion.