Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 5 / a

Table of contents
- Solution a

Solution

Let

D \subset R^{d}

be bounded, choose

R

with

D \subset B (0, R)

, and let

T

and

τ_{R}

be the respective exit times. Then

T \leq τ_{R}

. Since

∣ B_{t} ∣^{2} - d t

is a martingale, the optional sampling theorem for a supermartingale at

τ_{R} \land n

gives

d E_{x} (τ_{R} \land n) = E_{x} ∣ B_{τ_{R} \land n} ∣^{2} - ∣ x ∣^{2} \leq R^{2} - ∣ x ∣^{2} .

Monotone convergence theorem now gives

E_{x} T \leq E_{x} τ_{R} \leq \frac{R ^{2} - ∣ x ∣ ^{2}}{d} < \infty.

Solved by gpt-5.6-sol high.

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