Why is $(s,t) \mapsto \lvert t-s \rvert$ a control?

Question

On page 22 of Multidimensional Stochastic Processes as Rough Paths by Friz and Victoir, they state that a map $\omega \colon \Delta_T \to [0,\infty)$, where $\Delta_T = \{(s,t) : 0 \leq s \leq t \leq T\}$, is defined as superadditive if for all $s \leq t \leq u$ in $[0,T]$, $\omega(s,t) + \omega(t,u) \leq \omega(s,u)$. If $\omega$ is also continuous and zero on the diagonal, it is called a control. They then go on to saying that $(s,t) \mapsto \lvert t-s \rvert$ is an example of a control. But isn't this subadditive? How can it then be a control?

Answer 1

Notice how the control is defined on the simplex, so you have $s \le t\le u$ $$ \omega(s,t)=|t-s|=t-s $$ and $$ \omega(t,u)=u-t $$ so you have $$ \omega(s,t)+\omega(t,u)=\omega(s,u) $$ i.e. the function is additive on $\Delta_T$