Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 30K / d / ii / Solution Created 2026-09-24 Updated 2026-10-03
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Conversely, for a supermartingale define
(1)
The assumed equality of expectations gives
(2)
A nonnegative random variable of expectation zero vanishes almost surely, so
(3)
for every . Hence is a martingale. Together with part (i), this proves the constant-expectation supermartingale is a martingale criterion.
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