I'm having trouble figuring out how to solve this probability. I'm mostly confused about handling the dX(t) equation. Am I supposed to utilize Ito's Lemma with the dX(t) equation?
1 Answer
I would take a look here for a solution to this stochastic differential equation, for which we have the general solution
$$ ln \frac{S_t}{S_0} = (\mu - \frac{\sigma^2}{2})t + \sigma W_t $$
The key point in the reference is that
$$ \begin{align} d(\ln S_t) &= \frac{d S_t}{S_t} -\frac{1}{2}\frac{1}{S_t^2} dS_t dS_t \\ &= \frac{d S_t}{S_t} -\frac{1}{2} \frac{1}{S_t^2} \sigma^2 S_t^2 dt \end{align} $$
Think about rearranging the first equation and about the properties of a Brownian motion, specifically that $(Z_{t} - Z_s) \sim N(0, t - s)$
Also note that if $Y$ has a normal distribution then $X = e^Y$ has a lognormal distribution.
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$\begingroup$ You really shouldn't have completely solved the homework problem. But it was good of you to leave an error in the solution. $\endgroup$ Apr 21, 2016 at 0:00
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$\begingroup$ @MarkL.Stone Sorry was unaware of the rules regarding this / whether in fact this is homework? Is there guidelines somewhere related to this? $\endgroup$– mgilbertApr 21, 2016 at 0:31
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$\begingroup$ Look here stats.stackexchange.com/tags/self-study/info . It was linked in gung's comment above. $\endgroup$ Apr 21, 2016 at 0:37
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$\begingroup$ @MarkL.Stone My mistake, I have attempted to edit appropriately $\endgroup$– mgilbertApr 21, 2016 at 0:48
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$\begingroup$ Appreciate your effort, but you seemed to have left the most "intellectual" part completely solved. I suppose it's academic now, given access to edit history. $\endgroup$ Apr 21, 2016 at 0:50
[self-study]
tag & read its wiki. $\endgroup$