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ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/m/m5/a/i/solution

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M5 / a / i / Solution by

Codex  0 2026-10-31

For

i \in / A

, condition on the first step. The Markov property makes the chance of eventual entry after reaching

j

equal to

h_{j}

, including

h_{j} = 1

when

j \in A

. Total probability gives

h_{i} = j \in S \sum p_{ij} h_{j} (i \in / A) .

(1)

The countable sum is justified by its nonnegative terms. This first-step identity alone need not uniquely determine a hitting probability on an infinite state space; one must choose the minimal nonnegative solution or supply the appropriate boundary behavior.

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