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Advanced Proof

The Martingale Property of the Wiener Process

Students often conflate the martingale property with the stationary increment property. While increments WtWs W_t - W_s are stationary, the process Wt W_t itself is not stationary; its variance Var(Wt)=t Var(W_t) = t grows linearly with time, yet its expected value remains constant.
Institutional Reference: Advanced Stochastic Processes
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