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

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- Solution a

Solution

An

R^{d}

-valued process

(X_{t})_{t \geq 0}

is a Brownian motion started at

x

when:

$X_{0} = x$ almost surely;
its sample paths are almost surely continuous;
increments over disjoint time intervals are independent; and
for $0 \leq s < t$ , $X_{t} - X_{s}$ has the centered multivariate normal distribution with covariance matrix $(t - s) I_{d}$ .

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