Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 27J / b

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- Solution b

Solution

The scaled process starts at zero and has continuous paths. Disjoint time intervals map to disjoint intervals under

t \mapsto c^{2} t

, preserving independence of increments, and

\frac{B _{c^{2} t} - B _{c^{2} s}}{c} \sim N (0, \frac{c ^{2} ( t - s )}{c ^{2}}) = N (0, t - s) .

This holds for either sign of

c \neq = 0

. Hence

B_{t} = c^{- 1} B_{c^{2} t} is standard Brownian motion

.

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