# Solving Brownian Motion Probabilities

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? • Whether you use Ito's Lemma, or pray to the heavens above, one way or another you better solve (integrate) the stochastic differential equation to find the distribution of X(10). Apr 20, 2016 at 15:14
• that gave me an idea and was a very helpful start. thanks! Apr 20, 2016 at 15:26
• Please add the [self-study] tag & read its wiki. Apr 20, 2016 at 18:13
• (I see no reason that this needs to be closed.) Apr 20, 2016 at 18:14
• I'm curious since I've never worked with Ito's calculus. I know that the realization at time 10 is Gaussian, and I know that the covariance between realizations between in stationary Brownian motion. What's the relationship between the covariance and drift if there is any? Apr 20, 2016 at 23:53

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.

• You really shouldn't have completely solved the homework problem. But it was good of you to leave an error in the solution. Apr 21, 2016 at 0:00
• @MarkL.Stone Sorry was unaware of the rules regarding this / whether in fact this is homework? Is there guidelines somewhere related to this? Apr 21, 2016 at 0:31
• Look here stats.stackexchange.com/tags/self-study/info . It was linked in gung's comment above. Apr 21, 2016 at 0:37
• @MarkL.Stone My mistake, I have attempted to edit appropriately Apr 21, 2016 at 0:48
• 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. Apr 21, 2016 at 0:50