Properties and classic martingales of Brownian motion

State the definition of a standard Brownian motion, list its key properties, and name the standard martingales built from it.

$W_0=0$, continuous paths, independent increments with $W_t-W_s\sim N(0,t-s)$. Properties: Gaussian process with $\operatorname{Cov}(W_s,W_t)=\min(s,t)$, Markov, scale-invariant ($cW_{t/c^2}$ is again Brownian), quadratic variation $t$, nowhere differentiable, recurrent in one dimension. Classic martingales are $W_t$, $W_t^2-t$ and $\exp(\lambda W_t-\tfrac12\lambda^2t)$.