Suppose $X_t$ satisfy the following stochastic differential:
$dX_t = W_t^2 dW_t$
Let $Y_t = X_t^2$
Is $Y_t$ a martingale?
I first thought to find $X_t$ from the differential using Ito's differential and comparing coefficients then see if that was a martingale using the definition, then square it to see if $X_t^2$ is a martingale.
However, its impossible to express $X_t$ as a function of Brownian motion. Does this then infer that $X_t$ is not a martingale? If not, I'm not sure what other method to take?