Covariance of sums across three independent BMs

Question

Let $X$ , $Y$ and $Z$ be mutually independent standard Brownian motions. Compute the covariance between $X_1+Y_1$ and $Y_2+Z_2$ .

Short answer

$1$ . By bilinearity, every cross term between different, independent motions vanishes, leaving $\operatorname{Cov}(B_2(1),B_2(2))=\min(1,2)=1$ .

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Lessons behind this question

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

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