Given a non decreasing continuous function $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $f(0)=0$.
I want to find a continuous martingale $(X_t)_{t\ge 0}$ , such that it's quadratic variation is equal to $f$, i.e. $\langle X,X\rangle_t=f(t)$ for all $t\ge 0$.
Let $(W_t)_{t\ge 0}$ be standard brownian motion. I find $E[W_{f(t)}^2]=f(t)$. So I was considering to define $(X_t)_{t\ge0}$ with $X_t:=W_{f(t)}$.
Which just looks too simple to me. Is this actually right?