If $X_n$ is a martingale, is $X_n^2$ a submartingale?

Question

Let $(X_n,F_n)$ (n = 1,2,...) be a martingale. Is it true that $X_n^2$ is a submartingale?

Answer 1

Notice that $$\mathbb E\left[X_{n+1}^2\mid\mathcal F_n\right]=\mathbb E[(X_{n+1}-X_n+X_n)^2\mid\mathcal F_n]\geqslant 2\mathbb E\left[X_n(X_{n+1}-X_n)\mid\mathcal F_n\right] +X_n^2=X_n^2,$$ where the last "$=$" follows from the fact that $(X_n,\mathcal F_n)$ is a martingale.