Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 3 / a

Solution

Let

τ = T_{r} \land T_{R}

z \mapsto lo g ∣ z ∣

r < ∣ z ∣ < R

, so Itô formula shows that

lo g ∣ B_{t \land τ} ∣

lo g ∣ x ∣ = E_{x} [lo g ∣ B_{τ} ∣] = p lo g r + (1 - p) lo g R,

where

p = P_{x} (T_{r} < T_{R})

P_{x} (T_{r} < T_{R}) = \frac{l o g R - l o g ∣ x ∣}{l o g R - l o g r} .

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