Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 30L / a

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- Solution a

Solution

The increment of the stopped process is

M_{(n + 1) \land T} - M_{n \land T} = 1_{{T > n}} (M_{n + 1} - M_{n}) .

Because

T

is a stopping time,

{T > n} \in F_{n}

. Taking conditional expectation therefore gives zero. The stopped variables are adapted and integrable, so the stopped martingale in discrete time is a martingale.

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