 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2008/a/1/b/ii/solution

For

A \in F_{S}

, write

A \cap {S \leq T} \cap {T \leq n} = ⋃_{k = 0}^{n} [A \cap {S = k} \cap {k \leq T \leq n}] .

The first two factors together belong to

F_{k}

, since

A \cap {S = k}

is the difference of its intersections with

{S \leq k}

and

{S \leq k - 1}

. Also

{k \leq T \leq n} \in F_{n}

by the stopping time property. The filtration inclusion puts every term, and therefore the union, in

F_{n}

A \cap {S \leq T} \in F_{T} .

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