Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / c

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.

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