Consider a local martingale $(M_t)_{t\ge 0}$ with continuous paths and $\lim_{t\rightarrow\infty}[M]_t=\infty$ a.s.
I want to show, that
$$\lim_{t\rightarrow\infty}\frac{M_t}{[M]_t}=0\quad\text{a.s.}$$
I tried using fatou's lemma giving
\begin{align}
\liminf_{t\rightarrow\infty}E\bigg[\frac{M_t}{[M]_t}(1_{\{M_t<1\}}+1_{\{M_t\ge 1\}})\bigg]\le\liminf_{t\rightarrow\infty}E\bigg[\frac{M_t^2}{[M]_t}\bigg],
\end{align}
but I do not know how to go on further.
Another idea is to use borel-cantelli, since there are countable stopping times, I may be using solving this.
I would be grateful for any hint or help.
Best Answer
I believe that you need the further condition that $\langle M,M \rangle_\infty = \infty$ a.s., otherwise you can take $M$ to be something like Brownian motion stopped at $t=1$. Then you can use the method of time change: There exists a Brownian motion $B$ such that $M_t = B_{\langle M,M \rangle_t}$. Since you can show for Brownian motion that $\lim_{t \rightarrow \infty} \frac{B_t}{t} = 0$ a.s., the result still holds when $t$ is replaced by an increasing function that converges to $\infty$ a.s. so
$$\lim_{t \rightarrow \infty} \frac{M_t}{\langle M,M \rangle_t} = \lim_{t \rightarrow \infty} \frac{B_{\langle M,M \rangle_t}}{\langle M,M \rangle_t} = 0.$$