Brownian Motion with Drift and Scaling

Brownian motion
Distribution graph


The experiment consists of running the Brownian motion process \( \boldsymbol{X} = \{ X_s: s \in [0, \infty) \} \) with drift parameter \( \mu \) and scale parameter \(\sigma\) on the interval \( [0, t] \). The process \( \boldsymbol{X} \) can be constructed at \( X_s = \mu s + \sigma Z_s \) where \( \boldsymbol{Z} = \{Z_s: s \in [0, \infty)\} \) is standard Brownian motion. On each run, the path is shown in red in the graph on the left. The graph of the mean function \( m(t) = \mu t \) is shown in blue. The random variable of interest is the final position \( X_t \), which has the normal distribution with mean \( \mu t \) and standard deviation \( \sigma \sqrt{t} \).

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.