This?note of stochastic calculus covers Brownian motion and its basic properties.
1 Random Walks
1.1 Symmetric random walk
Let be a successive outcomes of the fair coin toss experiment (). Let’s define a random variable
and define , then we can construct following process , which is called a symmetric random walk.
With each toss, this process either steps up one unit or down one unit, and each of the two possibilities is equally likely.