\( \newcommand{\bs}{\boldsymbol} \)

### Reflected Brownian Motion

#### Description

Let \( \bs{X} = \{X_t: t \in [0, \infty)\} \) be standard Brownian motion, and let \( T_a = \min\{t > 0: X_t = a\} \), the first hitting time to a state \( a \gt 0 \). This app shows the process \( \bs{Y} = \{Y_s: s \in [0, \infty)\} \) on the interval \( [0, t] \), where
\[ Y_s = \begin{cases} X_s, & s \lt T_a \\ 2 a - X_s, & s \ge T_a \end{cases} \]
That is, the graph of \( \bs{X} \) is reflected in the line \( x = a \) after the process first hits \( a \). On each run, the path of \( \bs{X} \) is shown in red, while the reflected portion of the graph of \( \bs{Y} \) is shown in green (whenever \( \bs{X} \) hits \( a \)). The random variable of interest is \( Y_t \). Since \(\bs{Y}\) is also a standard Brownian motion, \( Y_t \) has the normal distribution with mean \( 0 \) and variance \( t \). The value of the variable is recorded in the first table. 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 \( t \) and \( a \) can be varied with input controls.