Finite quadratic variation leads to finite covariation

Question

I'm trying to prove that if two functions have finite quadratic variation then their covariation is finite. I've seen that $2|[X,Y]_{t}| \leq [X]_{t}+[Y]_{t}$ but I can't see how to get there. It seems like I want to prove $[X,Y]_{t} \leq min\{[X]_{t},[Y]_{t}\}$.

I was thinking maybe:

$$\sum_{0}^{n}(X_{k}-X_{k-1})(Y_{k}-Y_{k-1}) \leq min\{max\{X_{k}-X_{k-1}\}\sum_{0}^{n}(Y_{k}-Y_{k-1}), \\ max\{X_{k}-X_{k-1}\}\sum_{0}^{n}(Y_{k}-Y_{k-1})\} $$

But I couldn't really justify the argument to myself. Can anyone help?

Answer 1

Note that

$$[X-Y,X-Y] \geq 0$$

and since

$$[X-Y,X-Y] = [X,X]- 2 [X,Y] + [Y,Y]$$

this implies

$$2[X,Y] \leq [X,X] + [Y,Y].$$

Replacing $X$ by $-X$ and using $[-X,-X] = [X,X]$, we get

$$-2[X,Y] \leq [X,X] + [Y,Y].$$

Consequently,

$$2\big|[X,Y] \big| \leq [X,X]+[Y,Y].$$