 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/28k/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

E_{n}

be the holding time in state

n

. The variables are independent with

E_{n} \sim Exp (nλ)

, and the explosion time would be

T_{\infty} = \sum_{n \geq 1} E_{n} .

For

s > 0

E e^{- s T_{\infty}} = \prod_{n \geq 1} \frac{nλ}{nλ + s} .

This product is zero because

\sum_{n} lo g (1 + s / (nλ)) = \infty

. If

T_{\infty}

were finite with positive probability, the nonnegative variable

e^{- s T_{\infty}}

would have positive expectation. Therefore

T_{\infty} = \infty

almost surely, and the Yule process is nonexplosive.

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