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Stopped martingale in discrete time

For a discrete martingale and stopping time

T

,

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

The indicator is measurable at time

n

, so the stopped process is a martingale. If it is uniformly bounded and

T < \infty

almost surely, bounded convergence gives

E M_{T} = E M_{0}

directly.

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- Discounted symmetric random-walk exit transform Stopped martingale in discrete time 35

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