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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-211/6/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 6 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Apply the Itô formula to

ξ_{t} = V (t, S_{t})

. The backward equation cancels its drift, leaving

d ξ_{t} = (V_{t} + \frac{1}{2} a (S_{t})^{2} V_{SS}) (t, S_{t}) d t + a (S_{t}) V_{S} (t, S_{t}) d W_{t} = a (S_{t}) V_{S} (t, S_{t}) d W_{t} .

Bounded

a

and

V_{S}

make this Itô integral square-integrable on the finite horizon; in addition

V

itself is bounded. Hence

ξ

is a true martingale, not just a local martingale, and

ξ_{T} = g (S_{T})

. Taking conditional expectations gives

ξ_{t} = E [g (S_{T}) ∣ F_{t}] .

The stock is understood to be adapted to the stated Brownian filtration, with its initial value fixed there. This is the Feynman-Kac formula in the zero-potential, zero-drift case.

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