integrated Brownian Motion

Brownian motion
Distribution graph


The experiment consists of running the integrated Brownian motion process \( \boldsymbol{Y} = \{Y_s: s \in [0, \infty) \} \) on the interval \( [0, t] \), where \( Y_s = \int_0^s X_u \, du \) and \( \boldsymbol{X} = \{X_s: s \in [0, \infty)\} \) is standard Brownian motion. On each run, the path is shown in the graph on the left. The random variable of interest is the final position \( Y_t \), which has the normal distribution with mean 0 and standard deviation \( \sqrt{t^3 / 3} \).

On each run, the value of the variable is recorded in the first table, and the point \( (t, Y_t) \) 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 parameter \( t \) can be varied with the input control.