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

Set $A_n=\{{\sup }_{s\le t}|M_s^{(n)}|\ge 2^{-(n-1)/2}\}$. Part (a) gives ${\sum }_n\mathbb{P}(A_n)<\infty$. 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 satisfyThese deterministic bounds form a convergent geometric series. The finitely many earlier suprema are finite by continuity on the compact time interval. HenceThe series therefore converges absolutely and uniformly on $[0,t]$ by the Weierstrass M-test, and its sample-path limit is continuous.