# 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). – Mark L. Stone Apr 20 '16 at 15:14
• that gave me an idea and was a very helpful start. thanks! – Chuck Hotaling Apr 20 '16 at 15:26
• Please add the [self-study] tag & read its wiki. – gung Apr 20 '16 at 18:13
• (I see no reason that this needs to be closed.) – gung Apr 20 '16 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? – AdamO Apr 20 '16 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. – Mark L. Stone Apr 21 '16 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? – mgilbert Apr 21 '16 at 0:31
• Look here stats.stackexchange.com/tags/self-study/info . It was linked in gung's comment above. – Mark L. Stone Apr 21 '16 at 0:37
• @MarkL.Stone My mistake, I have attempted to edit appropriately – mgilbert Apr 21 '16 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. – Mark L. Stone Apr 21 '16 at 0:50