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With the standard one dimensional Wiener process $W_t$? Is $P(\omega: \exists \mbox{ finite} \lim\limits_{t\to \infty}W_t(\omega))=1$?

Could you help me!

Thank you in advance!

Kae
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    The probability should be zero. I haven't actually checked computations but why would you expect BM to settle near a point? –  Oct 26 '18 at 14:09
  • Thanks Zachary Selk. Here, I want to know the boundedness of $W_t(\omega)$? Since I see some documents say that $\lim\limits_{t\to\infty}\dfrac{W_t}{t}=0$ almost sure. So, how does $\lim\limits_{t\to\infty} W_t$? – Kae Oct 26 '18 at 14:31
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    $\limsup_{t \to \infty} W_t = \infty$ and $\liminf_{t \to \infty} W_t = - \infty$ almost surely, i.e. $W_t$ keeps oscillating. – saz Oct 26 '18 at 18:04
  • Thanks Saz so much! However, could you tell me more details or could you give me a document concerning this? Thank you in advance! – Kae Oct 27 '18 at 00:38

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