The Wiener process, also known as Brownian motion, models random continuous movement. It's characterized by continuous paths, meaning no sudden jumps. Its increments over non-overlapping time intervals are independent and normally distributed, with a mean of zero and variance proportional to the time interval's length. This means small changes are more likely than large ones. Although an idealization, it's fundamental in finance for modeling asset price fluctuations, where independent increments reflect the idea that past price movements don't predict future ones. Its mathematical tractability makes it crucial in stochastic calculus and various scientific fields.

Let's go step by step to justify the notation $dW= \xi \sqrt{dt}$ and why it leads to the result $dW^2=dt$

Step 1: Interpretation of dWdW as a Wiener Process

The Wiener process $W(t)$ describes Brownian motion and has the following properties:

1. Increment Distribution: The change $dW=W(t+dt)−W(t)$ follows a normal distribution:

   $$ dW \sim \mathcal{N}(0, dt) $$

   which means:

   $$ \mathbb{E}[dW] = 0, \quad \text{Var}(dW) = dt. $$

2. Independence: Non-overlapping increments of $W(t)$ are independent.

We want to understand why the notation $dW = \xi \sqrt{dt}$ is useful.

Step 2: Expressing $d W$ in Terms of a Normal Random Variable Since $d W$ is normally distributed with mean 0 and variance $d t$, we can express it as:

$$ d W=\xi \sqrt{d t} $$

where:

• $\xi \sim N (0,1)$ is a standard normal variable (zero mean, unit variance).
• The factor $\sqrt{d t}$ ensures that $d W$ has variance $d t$, since:

$$ E \left[(d W)^2\right]= E \left[\xi^2\right] d t=(1) d t=d t $$

This notation makes sense because in small time steps $d t$, the Wiener increment is proportional to $\sqrt{d t}$, which reflects the diffusive nature of Brownian motion.

Step 3: Why $d W^2=d t$ Holds Now, let's compute $d W^2$ :

$$ d W^2=(\xi \sqrt{d t})^2=\xi^2 d t $$

Taking the expectation:

$$ E \left[d W^2\right]= E \left[\xi^2\right] d t=(1) d t=d t $$