Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 3 / c

Solution

Put

θ_{t} = a t^{a - 1}

. When

a > 1/2

\int_{0}^{T} θ_{t}^{2} d t = \frac{a ^{2}}{2 a - 1} T^{2 a - 1} < \infty.

Q

\frac{d Q}{d P} = exp (- \int_{0}^{T} θ_{s} d W_{s} - \frac{1}{2} \int_{0}^{T} θ_{s}^{2} d s) .

W_{t}^{Q} = W_{t} + \int_{0}^{t} θ_{s} d s = W_{t} + t^{a} = S_{t}

Q

[0, T]

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