Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$
- $\mathcal M^2$ be the set of real-valued continuous $L^2(\operatorname P)$-bounded $\mathcal F$-martingales $X$ on $(\Omega,\mathcal A,\operatorname P)$ with $X_0=0$ equipped with $$\left\|X\right\|_{\mathcal M^2}^2:=\sup_{t\ge0}\left\|X_t\right\|_{L^2(\operatorname P)}^2\;\;\;\text{for }X\in\mathcal M^2$$
Now, let $X$ be a real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$. I want to show that the quadratic variation $[M]$ of $M$ has a nondecreasing modification.
In Theorem 15.5 of Foundations of Modern Probability, the author is proving the existence of $[M]$ with the mentioned property in the following way:
- Let $$\tau_k^n:=\inf\left\{t>\tau_{k-1}^n:\left|M_t-M_{\tau_{k-1}^n}\right|=\frac1{2^n}\right\}\;\;\;\text{for }k\in\mathbb N$$ with $\tau_0^n:=0$ for some $n\in\mathbb N$ and $$V_t^n:=\sum_{k\in\mathbb N}1_{\left(\tau_{k-1}^n,\:\tau_k^n\right]}(t)M_{\tau_{k-1}^n}\;\;\;\text{for }t\ge0$$
- Assume $M$ is bounded
- We can show that $$V^n\cdot M:=\sum_{k\in\mathbb N}M_{\tau_{k-1}^n}\left(M^{\tau_k^n}-M^{\tau_{k-1}^n}\right)$$ is an $\mathcal F$-martingale with $$\operatorname E\left[\left|\left(V\cdot M\right)_t\right|^2\right]\le\operatorname E\left[\left|M_t\right|^2\right]\;\;\;\text{for all }t\ge0\tag1$$
- Let $$Q^n:=\sum_{k\in\mathbb N}\left|M^{\tau_{k-1}^n}-M^{\tau_k^n}\right|^2\;\;\;\text{for }t\ge0$$
- By definition, $$M^2=2V^n\cdot M+Q^n\tag2$$
- Now, $$\left\|V^m\cdot M-V^n\cdot M\right\|_{\mathcal M^2}\le 2^{1-m}\left\|M\right\|_{\mathcal M^2}\;\;\;\text{for all }m\le n\tag3$$ and hence (since $\mathcal M^2$ is complete) $$\left\|V^n\cdot M-N\right\|_{\mathcal M^2}\xrightarrow{n\to\infty}0\tag4$$ for some $N\in\mathcal M^2$ ($\mathcal M^2$ is only a semi-normed space and hence $N$ is only unique up to indistinguishability)
- Now, the idea is to let $$\left[M\right]:=M^2-2N$$
Now, the author is arguing in a way that I absolutely don't understand: From $(4)$ he is concluding that $$\sup_{t\ge 0}\left(Q^n_t-[M]_t\right)\stackrel{\text{def}}=\sup_{t\ge 0}\left(\left(V^n\cdot M\right)_t-N_t\right)\xrightarrow{n\to\infty}0\;\;\;\text{in probability}\tag5$$ and, "in particular", $[M]$ is almost surely nondecreasing on $T:=\left\{\tau_k^n:k,n\in\mathbb N\right\}$.
While $(5)$ is clear, I don't see how he is able to conclude the nondecreasability on $T$ (and I don't understand why he is using convergence in probability at all, since we know that by $(4)$ the convergence in $(5)$ even holds in $L^2(\operatorname P)$).
Next he states the the monotonicity "extends by continuity to the closure $\overline T$ and since $[M]$ is constant on each interval in $\overline T^c$m $[M]$ is almost surely nondecreasing".
I understand nothing from this last conclusion. So, how can we rigorously show that $[M]$ has a nondecreasing modification?