### Geometric Brownian Motion

#### Description

The experiment consists of running the geometric Brownian motion process
\[ \boldsymbol{X} = \left\{X_s = \exp\left[\left(\mu - \frac{\sigma^2}{2}\right) s + \sigma Z_s)\right]: s \in [0, \infty) \right\} \]
on the interval \( [0, t] \), where \( \boldsymbol{Z} = \left\{Z_s: s \in [0, \infty)\right\} \) is standard Brownian motion. This random process is the solution of the stochastic differential equations
\[d X_t = \mu X_t \, dt + \sigma X_t \, dZ_t \]

On each run, the path is shown in the graph on the left in red. The graph of the mean function \( m(s) = e^{\mu s} \) is shown in blue. The random variable of interest is the final position \( X_t \), which has the lognormal distribution with parameters \( \left(\mu - \frac{\sigma^2}{2}\right) t \) and \( \sigma \sqrt{t} \).

On each run, the value of the variable is recorded in the first table, and the point \( (t, X_t) \) is shown as a red dot in the path graph on the left. The probability density function and moments, and the empirical density function and moments, are shown in the distribution graph on the right and given in the distribution table on the right. The parameters \( t \), \( \mu \), and \( \sigma \) can be varied with input controls, and the exponential parameter \( \mu - \frac{1}{2} \sigma^2 \) is also given.