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For $B_t$ Brownian Motion with drift $\mu<0$, I need to prove that the max value, $X = \max_{0 is finite almost surely, ie $P(X<\infty)=1$.

Now, I know that because the mean is negative, it will go more and more negative, and it is also a supermartingale. But I don't know how to prove almost surely...

Appreciate any hints.

2 Answers 2

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Hint: Try the strong law of large numbers. What does it say about $\lim_{t \to \infty} B_t/t$? What does this say about the sign of $B_t$ for large $t$?

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    Yes it never reaches + infinity. But the statement that it stays finite almost surely is not the correct way to express the result.2012-07-11
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Actually that is wrong in the sense that it is not finite in the limit. Brownian Motion with a negative drift will wander off to -∞ almost surely. It will tend to go down in a linear fashion with the slope equal to the drift parameter.

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    Yes, this is more than what I need to prove, I just need to prove that the maximal value is less than **positive** infinity. How do I do it?2012-07-11