Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 30K / a

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- Solution a

Solution

A process

(X_{n})_{n \geq 0}

is a martingale with respect to a filtration

(F_{n})_{n \geq 0}

when:

$X_{n}$ is $F_{n}$ -measurable for every $n$ , so the process is adapted;
$E ∣ X_{n} ∣ < \infty$ for every $n$ ;
for every $n \geq 0$ ,
$E [X_{n + 1} ∣ F_{n}] = X_{n} almost surely .$
(1)

Equivalently, every increment has conditional mean zero given the past.

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