Let $W$ be a standard, one-dimensional Brownian motion and $0 < T < \infty$. Show that
$$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
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chandu1729
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I was trying to use BDG inequality and Ito formula. But nothing worked out. I was able to prove its its weakest form i.e. the given integral (with out sup) converges to 0 in probability – chandu1729 Oct 18 '13 at 19:15
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Can you show where in the BDG and Ito you ran into trouble? – Oct 18 '13 at 19:19
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No, I was not able to progress. I just tried to use it. Any help is highly appreciated. This is a problem from the book by Karatzas and Shreve (BM and Stochastic Calculus). Thanks – chandu1729 Oct 18 '13 at 19:24