Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 2 / ii

Table of contents
- Solution ii

Solution

The stochastic process

W

starts from zero and has continuous paths. Its increments on disjoint intervals are independent, since the corresponding pairs of Brownian motion increments are independent, and

W_{t} - W_{s} \sim N (0, (ρ^{2} + 1 - ρ^{2}) (t - s)) = N (0, t - s) .

Thus $W$ is a standard Brownian motion, including $ρ = \pm 1$ . Here the intended filtration is the joint natural filtration, or another filtration relative to which the pair is a two-dimensional Brownian motion; independence of their path laws alone would not justify arbitrary extra information in a larger filtration.

By bilinearity of quadratic covariation and part (i),

⟨ W, B ⟩_{t} = ρ ⟨ B ⟩_{t} + 1 - ρ^{2} ⟨ B, B ⟩_{t} = ρt .

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