Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 30K / b

Table of contents
- Solution b

Solution

The increments of

Y_{n} = M_{n \land T}

satisfy

Y_{n} - Y_{n - 1} = 1_{{T \geq n}} (M_{n} - M_{n - 1}) .

Since

{T \geq n} = {T > n - 1} \in F_{n - 1}

, the indicator is predictable. Part (a), applied to this bounded integrand, shows that

Y

is a martingale. Equivalently, this is the stopped martingale theorem.

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