 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-2/26i/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Condition on the first jump, putting

h_{W_{0}} = 1

(λ + α) h_{C_{i}} = λ h_{C_{i + 1}} + α h_{W_{i}} (i \geq 0), (μ + β) h_{W_{i}} = μ h_{W_{i - 1}} + β h_{C_{i}} (i \geq 1) .

The relevant characterization is that a hitting probability is the minimal nonnegative harmonic extension of its target-boundary value. Thus among nonnegative solutions of these equations with

h_{W_{0}} = 1

, choose the componentwise least one. Other harmonic solutions need not be probabilities. This also identifies the solution without imposing an unjustified boundary condition at infinity.

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