Show that the limits (18.5) and (18.6) actually hold in \(L^{2}(\mathbb{P})\) and uniformly for \(T\) from compact sets.

To show that the limits hold uniformly, use Doob's maximal inequality. For the \(L^{2}\)-version of (18.6) show first that \(\mathbb{E}\left[\left(\sum_{j}\left(B_{t_{j}}-B_{t_{j-1}}\right)^{2}\right)^{4}\right]\) converges to \(c T^{4}\) for some constant \(c\).

Data From (18.5) And (18.6) 

Transcribed Image Text:

N g(B-)(B- B-1) 1=1 N 8(5)(1 - 1-1)2 1=1 P P 0 T T g(Bs) dBs (18.5) g(B) ds (18.6) 0