 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-25/2/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Multiply the representation in part (c) by

\int_{0}^{\infty} α_{u} d W_{u}

and take expectations. This Itô integral has mean zero, and the bilinear form of the Itô isometry gives

E [ϕ (W_{T}) \int_{0}^{\infty} α_{u} d W_{u}] = E \int_{0}^{\infty} β_{u} α_{u} d u .

Equating this with part (b), and writing both ordinary integrals with the same time variable, yields

E \int_{0}^{\infty} (β_{u} - ϕ^{'} (W_{T}) 1_{{u \leq T}}) α_{u} d u = 0.

All terms are integrable by the Cauchy-Schwarz inequality, the assumed square integrability of

α

, and boundedness of

ϕ^{'}

. This is an orthogonality statement against predictable processes; its second term need not itself be predictable.

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