Show that the limits (18.5) and (18.6) actually hold in (L^{2}(mathbb{P})) and uniformly for (T) from compact
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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)
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Related Book For
Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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