Symmetry of Brownian Motion

Stochastic Processes

Brownian Motion

Easy

For standard Brownian motion, what is

P (B_{1} > 0)

Hint 1

$B_{1}$ is a continuous random variable symmetric about zero.

Hint 2

$B_{1} \sim N (0, 1)$ , and the standard normal is symmetric about $0$ .

Show worked solution

1. Identify the distribution
$B_{1} \sim N (0, 1)$ , which is symmetric around $0$ and has no atom at $0$ .
2. Apply symmetry
By symmetry, $P (B_{1} > 0) = P (B_{1} < 0) = 1/2$ .
$P (B_{1} > 0) = \frac{1}{2}$

Answer: 1/2

1. Identify the distribution