Commuting supremum and expectation

Question

Suppose to have random variables $X_t \geq 0$ for every $t\in [0,1]$. Suppose also that $E[X_t] < \infty$ uniformly for every $t \in [0,1]$ (which implies $\sup_{t \in [0,1]} E[X_t] < \infty$). Can I conclude that $$ E[\sup_{t \in [0,1]} X_t] < \infty ? $$ Or, more in general, under which conditions can I say that there exists $C > 0$ such that $$ E[\sup_{t \in [0,1]} X_t] \leq C \sup_{t \in [0,1]} E[X_t] ? $$

Answer 1

No. Even worse, it is possible that under your assumptions $\sup_{0\leq t\leq 1}X_t=\infty$ with positive probability. Consider the following example:

Let $X_0\sim\mathcal N(0,1)$ and define $$X_t=\begin{cases}\frac{1}{1-t}X_0, & t\in [0,1) \\ 0, & t=1\end{cases}$$ Then clearly $\lim_{t\nearrow 1}X_t=\text{sign}(X_0)\cdot\infty$.
Therefore $$\mathbb E\sup_{0\leq t\leq 1}X_t=\mathbb P[X_0>0]\cdot\mathbb E\bigg[\sup_{0\leq t\leq 1}X_t\bigg|X_0>0\bigg]+ \mathbb P[X_0<0]\cdot\mathbb E\bigg[\sup_{0\leq t\leq 1}X_t\bigg|X_0<0\bigg]=+\infty$$

Furthermore, an inequality as you describe it, i.e. $\mathbb E\sup_{0\leq t\leq 1}X_t\leq C \cdot\sup_{0\leq t\leq 1}\mathbb EX_t$, is unlikely to hold for a large class of processes satisfying your conditions. Simply consider a zero-mean process, then such an inequality would imply that $\sup_{0\leq t\leq 1}X_t=0$ which together with the condition $\mathbb EX_t=0$ implies $X_t=0$ almost-surely.

Though perhaps an interesting class where similar bounds can be achieved are Submartingales. In particular you may want to take a look at Doob's Martingale Inequality.