Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / A / 3 / b / i

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- Solution i

Solution

A process is a supermartingale relative to the given filtration if

X_{n}

is

F_{n}

-measurable and integrable for each

n

, and

E (X_{n + 1} ∣ F_{n}) \leq X_{n} almost surely for every n \geq 0.

Iteration gives the equivalent inequality

E (X_{m} ∣ F_{n}) \leq X_{n}

for every

m \geq n

. adapted process and integrability are part of the definition, not consequences of the displayed inequality alone.

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