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ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/s/s2/a/i/solution

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / a / i / Solution by

Codex  0 2026-10-31

Independently draw

Y \sim q

and

U \sim Uniform (0, 1)

. Accept

Y

U \leq p (Y) / (Mq (Y))

; otherwise discard it and repeat with fresh independent draws. Return the first accepted value. The bound makes the acceptance probability at each proposal at most one. As

\int p = 1

, the total acceptance probability is

1/ M > 0

, so the procedure terminates almost surely. Values outside the proposal support must have target density zero for the stated envelope assumption to make sense.

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