Martingale Property of Brownian Motion
Let $W_t$ be a standard Brownian motion. Define the process $M_t = W_t^2 - t$.
Show that $M_t$ is a martingale with respect to the natural filtration $\mathcal{F}_s = \sigma(W_u : u \leq s)$. That is, verify that $E[M_t \mid \mathcal{F}_s] = M_s$ for all $0 \leq s < t$.
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