This?note of stochastic calculus covers Brownian motion and its basic properties.

## 1 Random Walks

### 1.1 Symmetric random walk

#### 1.1.1 Definition

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.