Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 4 / b

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- Solution b

Solution

The paths of

U X_{t}

are continuous, and linear transformation preserves independence of increments. Moreover,

U (X_{t} - X_{s}) \sim N (0, (t - s) U I_{d} U^{T}) = N (0, (t - s) I_{d})

because

U

is an orthogonal matrix. Thus all defining properties are preserved, proving the orthogonal invariance of Brownian motion:

(U X_{t})_{t \geq 0} is Brownian motion in R^{d} .

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