Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 4 / b / Solution

Enlarge the space by an independent Brownian motion

W

B_{t} = \int_{0}^{t} 1_{{A_{s} > 0}} A_{s}^{- 1/2} d X_{s} + \int_{0}^{t} 1_{{A_{s} = 0}} d W_{s} .

The two terms have zero cross-variation and

[B]_{t} = \int_{0}^{t} 1_{{A_{s} > 0}} d s + \int_{0}^{t} 1_{{A_{s} = 0}} d s = t .

The Lévy characterization of Brownian motion makes

B

a Brownian motion. The residual

\int 1_{{A = 0}} d X

has zero quadratic variation and is therefore constant, so

X_{t} - X_{0} = \int_{0}^{t} A_{s}^{1/2} d B_{s} .

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