Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 33 / 5 / a

Solution

Put

τ = T_{r} \land T_{R}

. First establish almost sure finiteness. The Itô formula gives the martingale

∣ B_{t} ∣^{2} - 2 t

for planar Brownian motion. Stop it at

t \land T_{R}

. The stopped path has norm at most

R

, so

2 E (t \land T_{R}) = E ∣ B_{t \land T_{R}} ∣^{2} - ∣ x ∣^{2} \leq R^{2} - ∣ x ∣^{2} .

E T_{R} < \infty

, and hence

τ < \infty

h (y) = lo g ∣ y ∣

0

h^{''} (ρ) + h^{'} (ρ) / ρ = - ρ^{- 2} + ρ^{- 2} = 0

. The Itô formula shows

h (B_{t \land τ})

is a local martingale. It is bounded between

lo g r

and

lo g R

, so it is a true martingale and passage to

τ

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

and using continuity at exit gives

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

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

The stopping argument stays away from the singularity of the logarithm until the calculation is complete.

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