Supremum of random variable and function

Question

I have a silly question that, however, I'm not sure of what's the answer. If I have $$\sup_{0 \le s \le t}\big(X_s-s\big)$$ where $t \ge s$ then it correspond to $\sup_{0 \le s \le t}(X_s)-s$ or to $\sup_{0 \le s \le t}(X_s)-0$?? So the supremum impact also on the deterministic function or not?

Answer 1

In general we have that the following inequality holds: $$\sup_{0 \le s \le t}\big(X_s-Y_s\big)\leq \sup_{0 \le s \le t}X_s +\sup_{0 \le s \le t}(-Y_s)=\sup_{0 \le s \le t}X_s -\inf_{0 \le s \le t}Y_s.$$ Moreover both $\sup_{0 \le s \le t}X_s$ and $\inf_{0 \le s \le t}Y_s$ depend on $t$ (not on $s$). For $Y_s=s$, $\inf_{0 \le s \le t}Y_s=0$.