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Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:

If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \rightarrow \infty$ as $t \rightarrow \infty$

If $\mu < \sigma^2/2$, then $X_t \rightarrow 0$ as $t \rightarrow \infty$

If $\mu = \sigma^2/2$, then $X_t$ has no limit as $t \rightarrow \infty$

While this makes sense, how would the proof look like for this property? I'm not really sure how to approach it at the moment. Any help is appreciated.

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For any $\alpha > 0$, \begin{align*} \lim_{t\rightarrow\infty}P\left(e^{\big(u-\frac{\sigma^2}{2}\big) t +\sigma W_t} > \alpha \right) &= \lim_{t\rightarrow\infty}P\left(\big(u-\frac{\sigma^2}{2}\big) t +\sigma W_t > \ln \alpha \right)\\ &=\lim_{t\rightarrow\infty}P\left(\frac{W_t}{\sqrt{t}} > \frac{\ln\alpha- \big(u-\frac{\sigma^2}{2}\big) t }{\sigma \sqrt{t}} \right)\\ &=\lim_{t\rightarrow\infty}\Phi\left(\frac{\big(u-\frac{\sigma^2}{2}\big) t -\ln\alpha}{\sigma \sqrt{t}} \right)\\ &= \begin{cases} 1, &\mbox{ if } u>\frac{\sigma^2}{2},\\ \frac{1}{2}, &\mbox{ if } u=\frac{\sigma^2}{2},\\ 0, &\mbox{ if } u < \frac{\sigma^2}{2}. \end{cases} \end{align*} The conclusion now follows immediately.

Edit based on comments below.

Let $X_t = e^{\big(u-\frac{\sigma^2}{2}\big) t +\sigma W_t}$. Note that \begin{align*} \left(\omega:\, \lim_{t\rightarrow \infty} X_t = \infty \right) &= \cap_{m=1}^{\infty}\cup_{n=1}^{\infty} \cap_{t\ge n}(\omega:\,X_t > m). \end{align*} The almost surely convergence, for the case where $u>\frac{\sigma^2}{2}$, follows immediately.

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  • $\begingroup$ This answer was flagged with the following comment from @antonzm 'It seems that what is proven here is different from the statement that is required. The general statement is, for instance, that 𝑋(𝑑)β†’βˆž a.s. as π‘‘β†’βˆž when 𝑒>𝜎22, however, what is shown in here is that 𝑋(𝑑)β†’βˆž in probability as π‘‘β†’βˆž. To show the former result, an argument with the law of iterated expectations must be done to argue that the 𝑑 term dominates the π‘Šπ‘‘ term in the limit.' I have copied the flag contents here to allow Gordon chance of rebuttal or amendment. $\endgroup$
    – Attack68
    Dec 4 '19 at 17:39
  • $\begingroup$ @Attack68: I will have a look. In fact, antonzm can actually make comments here directly. $\endgroup$
    – Gordon
    Dec 4 '19 at 18:25
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Write $X_t = \exp(\mu B_t + (\mu - \frac {\sigma^2}{2})t )$. To prove the statement you can use the law of large numbers for Brownian motion which states that $\lim_{t \to \infty} \frac {B_t}{t} = 0$. Then rewrite $X_t$ as $$X_t = \exp(t (\mu \frac {B_t}{t} + (\mu - \frac{\sigma^2}{2})).$$ Using these two properties, you can analyze the convergence behaviour of $X_t$.

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