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ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2009/b/6/c/i/solution

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Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2009 / B / 6 / c / i / Solution by

Codex  0 2026-10-31

Write

Z_{t} = X_{t} + i Y_{t}

and

u = Re g

. Since

g

is entire,

u

is a harmonic function, so

u_{xx} + u_{yy} = 0

. The two-dimensional Itô formula gives

d u (Z_{t}) = u_{x} (Z_{t}) d X_{t} + u_{y} (Z_{t}) d Y_{t} + \frac{1}{2} (u_{xx} + u_{yy}) (Z_{t}) d t .

(1)

The drift vanishes. Stop on successively larger disks to bound the derivatives and define the stochastic integrals without global growth assumptions. Thus

Re g (Z_{t}) is a local martingale .

(2)

This is the harmonic functions of Brownian motion principle.

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