 Solution

For $s<t$, write $X_t=X_s+Y$, where the increment $Y=X_t-X_s$ is independent of the natural filtration at time $s$, has mean zero, and has variance $(t-s){\sigma }^2$. HenceIt follows that $M_t=X_t^2-t{\sigma }^2$ is a martingale, as asserted by the centered square-integrable Lévy martingale identity.

Solved by gpt-5.6-sol high.