Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 4 / b

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- Solution b

Solution

The paths of

U X_{t}

are continuous and start at zero. Its increments are independent because they are deterministic functions of the independent increments of

X

. They are centered Gaussian, and orthogonality gives

Cov (U (X_{t} - X_{s})) = U ((t - s) I_{d}) U^{T} = (t - s) I_{d} .

Thus

U X

is Brownian motion. This is the orthogonal invariance of Brownian motion.

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