Let $W=(W(t))_{t\geq 0}$ be a Brownian motion. Consider the random variable $$Y(t):=\sup_{1\leq s\leq t}[W(s)-W(s-1)],$$ for some fixed instant $t\geq 0$. I am interested in this $Y(t).$
Could anyone get an estimate for the first (in absolute value) or second moment of $Y(t)$, i.e., $\mathbb E|Y(t)|$ or $\mathbb E(Y(t))^2$? I haven't succeeded thus far.*
*Note: An inequality such as $$|Y(t)|\leq \sup_{1\leq s\leq t}|W(s)|+\sup_{1\leq s\leq t}|W(s-1)|$$ will result into a too blunt estimate; I would like a sharper estimate.