Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$
Let $X^n$ be a real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ with $X_0^n=0$ for $n\in\mathbb N$. I want to show that$^1$ $$\sup_{t\ge0}\left|X_t^n\right|\xrightarrow{n\to\infty}0\text{ in probability}\Leftrightarrow\lim_{t\to\infty}[X^n]_t\xrightarrow{n\to\infty}0\text{ in probability }\tag3\;.$$
How can we prove $(3)$?
For the first direction "$\Rightarrow$": Let $\varepsilon>0$. By assumption, $$\operatorname P\left[\sup_{t\ge0}\left|X^n_t\right|>\varepsilon\right]\xrightarrow{n\to\infty}0\tag4\;.$$ Let $$Y^n:=(X^n)^2-[X^n]$$ and $$\tau_n:=\inf\left\{t\ge0:|X^n_t|>\varepsilon\right\}$$ for $n\in\mathbb N$. By $(2)$, we obtain $$\operatorname E\left[[X^n]_t^{\tau_n}\right]\le\varepsilon^2\;\;\;\text{for all }t\ge0\tag5\;,$$ but I don't know how I need to proceed.
(By definition, the limit as $t\to\infty$ on the right-hand side of $(3)$ is equal to the supremum over $t\ge0$. Am I missing something or isn't it possible that this supremum is $\infty$?)
$^1$ If $X$ is a real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$, there is a real-valued $\mathcal F$-adapted stochastic process $[X]$ on $(\Omega,\mathcal A,\operatorname P)$, unique up to indistinguishability, with
- $[X]_0=0$
- $[X]$ is continuous
- $[X]$ is of locally bounded variation
- $X^2-[X]$ is a local $\mathcal F$-martingale
- $[X]$ is nondecreasing
If $(\sigma_n)_{n\in\mathbb N}$ is a localizing sequence for $X$, then $(\sigma_n\wedge n)_{n\in\mathbb N}$ is a localizing sequence for both $X$ and $X^2-[X]$. If $X$ is an $\mathcal F$-martingale, then $X^2-[X]$ is an $\mathcal F$-martingale. If $\tau$ is an $\mathcal F$-stopping time on $(\Omega,\mathcal A)$, then $$[X^\tau]=[X]^\tau\tag1\;.$$ If $X_0=0$, then $$\operatorname E\left[[X]_t\right]\le\operatorname E\left[\sup_{s\in[0,\:t]}\left|X_s\right|^2\right]\le 4\operatorname E\left[[X]_t\right]\;\;\;\text{for all }t\ge0\tag2\;.$$