Determining if process $Z$ is a martingale.

Question

I have to determine whether a process $Z$ defined by $Z_0 = 0$ and $Z_t = W_t^5 - 10 \int_0^t W_u^3 du$ is a martingale. Where $W_t$ is a standard Brownian motion.

I thought the best way to start was to determine $dZ_t$ and see what we get. So, I first used Ito's lemma to compute the first term. After some work I got $5W_t^4dW_t + 10 W_t^3dt$.

Then I wanted to compute $10 \int_0^t W_u^3 du$, however I didn't know how to compute it. Maybe I can use Ito's lemma again, but I don't know how to do it.

Could someone help me?

Answer 1

This is just some notational confusion. Remember that $dY_t = b_t dW_t$ is just notation for $Y_t = \int_0^t b_u dW_u$. In this problem, $Y_t := \int_0^t W_u^3dW_u$ is equivalent to $dY_t = W_t^3 dW_t$.