Let

- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
- $X$ be an almost surely continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$

It's a classical result that there is an $\mathcal F$-adapted stochastic process $\langle X\rangle$ on $(\Omega,\mathcal A,\operatorname P)$ with

- $\operatorname P\left[\langle X\rangle_0=0\right]=1$
- $\langle X\rangle$ is almost surely continuous
- $\langle X\rangle$ is almost surely increasing
- $|X|^2-\langle X\rangle$ is an almost surely continuous local $\mathcal F$-martingale

$\langle X\rangle$ is unique up to indistinguishability and is called **quadratic variation** of $X$.

The question is: Do we need to call it the quadratic variation *with respect to* $\mathcal F$? In other words: If $(\mathcal G_t)_{t\ge 0}$ is another complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ such that $X$ is a local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ and $\langle X\rangle'$ is the quadratic variation of $X$ with respect to $\mathcal G$, can we conclude that $\langle X\rangle$ and $\langle X\rangle'$ are indistinguishable?