4 edited title

# Limit of a Wiener integral \lim_{\alpha\rightarrow\infty}

3 deleted 5 characters in body; edited title

# Limit of a Wiener integral \lim_{\alpha\rightarrow\infty}

How to show that

$$\lim_{\alpha \rightarrow \infty } infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$ text{a.e.}$$where \left (B_s \right)_{s\geq 0} is a real standard brownian motion starting from zero ? I'd like to have some ideas to deal with this problem. After all, I'll show some solutions that I propose and somme colegues also but that i believe be all wrong. (I just don't show know to don't interffer in your ideas. Thank you all. 2 added 1 characters in body How to show that$$ \lim {\alpha lim_{\alpha \rightarrow \infty } \sup{t sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} 

where $\left (B_s \right)_{s\geq 0}$ is a real standard brownian motion starting from zero ?

I'd like to have some ideas to deal with this problem. After all, I'll show some solutions that I propose and somme colegues also but that i believe be all wrong. (I just don't show know to don't interffer in your ideas.

Thank you all.

1