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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-34/3/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 3 / a / Solution by

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

For

h (x, y) = \frac{1}{2} lo g (x^{2} + y^{2})

away from the origin,

\nabla h = \frac{( x , y )}{x ^{2} + y ^{2}}, h_{xx} = \frac{y ^{2} - x ^{2}}{( x ^{2} + y ^{2} ) ^{2}}, h_{yy} = \frac{x ^{2} - y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}, Δ h = 0.

(1)

The Itô formula for planar Brownian motion therefore has no drift term before the radius reaches

r

. Since the initial logarithm is zero,

lo g ∣ B_{t \land S_{r}} ∣ = \int_{0}^{t \land S_{r}} \frac{B _{s}^{1} d B _{s}^{1} + B _{s}^{2} d B _{s}^{2}}{∣ B _{s} ∣ ^{2}} .

(2)

The integrand's squared norm is

∣ B_{s} ∣^{- 2} \leq r^{- 2}

on this stopped interval, so this is a square-integrable martingale on every finite horizon, in particular a local martingale. One may first stop in an outer annulus to apply the formula on a compact smooth domain and then remove that localization.

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