Show that a certain functional of Brownian motion is a martingale

Question

Question: Show that $(W^2_{t}-t)^2 - 4 \int_{0}^{t} W^2_{u} du$ is a martingale.

I understand how to show that $(W^2_{t}-t)$ is a martingale, and I know that $4 \int_{0}^{t} W^2_{u} du$ is the quadratic variation of $(W^2_{t}-t)$, but I don't know how to use that to prove the first process is a martingale.

Answer 1

For any ($L^2$-)martingale $(M_t)_{t \geq 0}$ the quadratic variation $(\langle M \rangle_t)_{t \geq 0}$ is defined such that

$$M_t^2 - \langle M \rangle_t$$

is a martingale. Apply this to $M_t = W_t^2-t$. (There is no need for Itô's formula!)