If $X$ and $Y$ are local martingales, then $[X,Y]\le[X][Y]$

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
• $X$ and $Y$ be almost surely continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$

I want to show that $^1$ $$[X,Y]\le[X][Y]\;\;\;\text{almost surely}\;.\tag1$$

The idea is pretty clear to me:

• Let $t\ge s\ge0$
• By 5. - 7. below, we obtain \begin{equation}\begin{split}0&\le[X+\lambda Y]_t-[X+\lambda Y]_s\\&=([X]_t-[X]_s)+2\lambda([X,Y]_t-[X,Y]_s)+\lambda^2([Y]_t-[Y]_s)\end{split}\tag2\end{equation} almost surely for all $\lambda\in\mathbb R$ (I've found a proof where the author states that $(2)$ holds almost surely for all $\lambda\in\mathbb Q$ and can be "extended by continuity in $\lambda$" to all $\lambda\in\mathbb R$. I have no idea why such an argumentation is necessary and would be interested in any comment targeting this issue)
• Minimizing the right-hand side of $(2)$ with respect to $\lambda$ yields $$0\le[X]_t-[X]_s-\frac{\left|[X,Y]_t-[X,Y]_s\right|^2}{[Y]_t-[Y]_s}\tag3$$ and hence $$\left|[X,Y]_t-[X,Y]_s\right|^2\le([X]_t-[X]_s)([Y]_t-[Y]_s)\tag4$$

(a) My first problem is that $(2)$ holds only on the complement of a $\operatorname P$-null set which depends on $\lambda$. So, it's not clear to me how we need to aruge that $(3)$ holds almost surely. (b) My second problem is that, if we know that $(3)$ holds almost surely for all $t\ge s\ge0$, we need to argue that we can choose a common $\operatorname P$-null set for all $(s,t)$.

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$^1$ If $M$ and $N$ are almost surely continuous local $\mathcal F$-martingales on $(\Omega,\mathcal A,\operatorname P)$, then there is a $\mathcal F$-adapted stochastic process $[M,N]$ on $(\Omega,\mathcal A,\operatorname P)$ with

1. $[M,N]=0$
2. $[M,N]$ is continuous
3. $[M,N]$ is of locally bounded variation
4. $MN-[M,N]$ is a local $\mathcal F$-martingale

$[M,N]$ is unique up to indistinguishability. Moreover, there is a continuous and nondecreasing $\mathcal F$-adapted stochastic process $[M]$ of locally bounded variation on $(\Omega,\mathcal A,\operatorname P)$ with $[M]_0=0$. $[M]$ is unique up to indistinguishability. We can show that

5. $[M]=[M,M]$ almost surely
6. $[M,N]=[N,M]$ almost surely
7. If $Z$ is an almost surely continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$, then $$[\lambda M+N,Z]=\lambda[M,Z]$$ almost surely for all $\lambda\in\mathbb R$

Answer 1

To answer your question about (2).

We may think in this way: There exists $\Omega_{0}$ with $P(\Omega_{0})=1$ such that for each $\omega\in\Omega_{0}$ and for each $\lambda\in\mathbb{Q}$, $$ \left([X]_{t}-[X]_{s}\right)(\omega)+2\lambda\left([X,Y]_{t}-[X,Y]_{s}\right)(\omega)+\lambda^{2}\left([Y]_{t}-[Y]_{s}\right)(\omega)\geq0. $$ If $\lambda_{0}\in\mathbb{Q}^{c}$, we choose a sequence $(\lambda_{n})$ in $\mathbb{Q}$ such that $\lambda_{n}\rightarrow\lambda_{0}$. By putting $\lambda=\lambda_{n}$ in above and taking limit, we have $$ \left([X]_{t}-[X]_{s}\right)(\omega)+2\lambda_{0}\left([X,Y]_{t}-[X,Y]_{s}\right)(\omega)+\lambda_{0}^{2}\left([Y]_{t}-[Y]_{s}\right)(\omega)\geq0. $$ The key point here is that terms $\left([X]_{t}-[X]_{s}\right)(\omega)$, $\left([X,Y]_{t}-[X,Y]_{s}\right)(\omega),$ and $\left([Y]_{t}-[Y]_{s}\right)(\omega)$ are real numbers and will not be affected by $\lambda$.

Hence we have: For all $\omega\in\Omega_{0}$ and $\lambda\in\mathbb{R}$, $$ \left([X]_{t}-[X]_{s}\right)(\omega)+2\lambda\left([X,Y]_{t}-[X,Y]_{s}\right)(\omega)+\lambda^{2}\left([Y]_{t}-[Y]_{s}\right)(\omega)\geq0 $$

Remark: To yield inequality (3), we do not need to consider terms $[X+\lambda Y]_s$ and $[X+\lambda Y]_t$.