Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 30K / c / i

Table of contents
- Solution i

Solution

Assume (i), and let

τ

be a stopping time. The increments of the stopped process satisfy

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

Because

{τ > n} \in F_{n}

, taking conditional expectation given

F_{n}

makes the right-hand side zero. Integrability follows from

M_{n \land τ} = M_{0} + \sum_{k = 1}^{n} 1_{{τ \geq k}} (M_{k} - M_{k - 1}),

a finite sum of integrable random variables. Hence

M^{τ}

is a martingale, proving (ii).

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