 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/a/3/b/ii/solution

T : Ω \to N_{0} \cup {\infty}

such that

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

for every

n

. The convention is

in f \emptyset = \infty

For the hitting time in question,

{T_{k} \leq n} = ⋃_{j = 0}^{n} {S_{j} = k} .

Each event in the union is in

F_{j} \subseteq F_{n}

because

S_{j}

is adapted. This proves $T_{k}$ is a stopping time for every positive integer

k

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