$\langle B_t \rangle = t$. Quadratic variation of $B_t$

Question

We certainly know that $\langle B_t \rangle = t$. And I also know how to prove it using the definition of quadratic variation.

What I wanted to know is if there is also an alternative proof that make us of Ito's Formula. Can someone help?

Answer 1

By Doob-Meyer Decomposition theorem we know that for every continuous local martingale (starting at zero) $M,$ the quadratic variation process $[M]$ is the unique adapted continuous increasing process such that $M^2-[M]$ is a continuous local martingale starting at zero. We know that a B.M. is a continuous local martingale. So now, using Ito's lemma we can prove that $\{ B_t^2 -t \}$ is a martingale and therefore, $[B_t]=t.$

(But you can prove that $\{ B_t^2 -t \}$ is a martingale without using Ito' lemma)