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Let $b: \mathbb{R} \rightarrow \mathbb{R}$ be a Lipschitz-continuous function and let $X_t$ be a real valued stochastic process satisfying the stochastic differential equation $dX_t= b(X_t) dt+ dB_t$, $X_0=x$. Prove that for any $M> 0$, $t> 0$ and $x \in \mathbb{R}$ we have that $P(X_t \geq M)>0$ but in the case that $b(x)= \alpha$ for some $\alpha <0$ we have that $P(\lim_{t \rightarrow \infty} X_t= - \infty)=1$.

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    $\begingroup$ What do you know? What did you try? Where are you stuck? (Sounds like homework, is it homework?) $\endgroup$ – Did Jan 8 '12 at 22:32
  • $\begingroup$ I believe the first part follows from Girsanov's theorem. For the second part, you can solve the SDE explicitly, and then the law of iterated logrithms works. $\endgroup$ – Aaron Jan 9 '12 at 9:37

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