Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 3 / c / i

Table of contents
- Solution i

Solution

Let

Z_{t} = g_{t} (z) - U_{t}

. For

SLE_{4}

,

d U_{t} = 2 d B_{t}

, and the Chordal Loewner equation gives

d Z_{t} = \frac{2}{Z _{t}} d t - 2 d B_{t}, d ⟨ Z ⟩_{t} = 4 d t .

The complex Itô formula yields

d lo g Z_{t} = \frac{1}{Z _{t}} d Z_{t} - \frac{1}{2 Z _{t}^{2}} d ⟨ Z ⟩_{t} = - \frac{2}{Z _{t}} d B_{t} .

Therefore

lo g (g_{t} (z) - U_{t}) is a continuous local martingale .

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