Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 1 / a / ii

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- Solution ii

Solution

Both variables are centered. The Itô isometry in its bilinear form gives

Cov (B_{t}, β_{t}) = E [(\int_{0}^{t} sign (β_{s}) d β_{s}) (\int_{0}^{t} 1 d β_{s})] = \int_{0}^{t} E [sign (β_{s})] d s = 0,

where the last equality follows because a centered Gaussian distribution is a symmetric probability distribution. Thus

B_{t}

and

β_{t}

are uncorrelated random variables.

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