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ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2011/2/b/solution

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Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 2 / b / Solution by

Codex  0 2026-10-31

A_{n} = {sup_{s \leq t} ∣ M_{s}^{(n)} ∣ \geq 2^{- (n - 1) /2}}

. Part (a) gives

\sum_{n} P (A_{n}) < \infty

. The Borel-Cantelli first lemma implies that, almost surely, only finitely many

A_{n}

occur. On the complementary full-probability event, all sufficiently late terms satisfy

sup_{s \leq t} ∣ M_{s}^{(n)} ∣ < 2^{- (n - 1) /2} .

(1)

These deterministic bounds form a convergent geometric series. The finitely many earlier suprema are finite by continuity on the compact time interval. Hence

n \sum s \leq t sup ∣ M_{s}^{(n)} ∣ < \infty almost surely .

(2)

The series therefore converges absolutely and uniformly on

[0, t]

by the Weierstrass M-test, and its sample-path limit is continuous.

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