Covariance of Brownian motion at times one and two

Question

For a standard Brownian motion $W$ , compute the covariance between $W_1$ and $W_2$ .

Short answer

$1$ . Write $W_2=W_1+(W_2-W_1)$ , where the increment is independent of $W_1$ with mean zero. Then $\operatorname{Cov}(W_1,W_2)=\operatorname{Var}(W_1)=1$ ; in general $\operatorname{Cov}(W_s,W_t)=\min(s,t)$ .

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Lessons behind this question

Itô's Formula
Covariance and Correlation
The Four Faces of Brownian Motion

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