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## What is Brownian motion?

**Motivation**:
Brownian motion is also called *Wiener Process* is the erratic random movement of
microscopic particles in a fluid, as a result of continuous bombardment from
molecules of the surrounding medium. It was first observed by botanist Robert
Brown in 1827. Norbert Wiener showed the mathematical model of Brownian motion
(and therefore Brownian motion is also called Wiener Process). Albert Einstein
wrote a paper in 1905 about Brownian motion relation to Avogadro number, Jean
Perrin got a Nobel Prize in 1926 for his experimental results in Brownian
motion and in 1997 Merton and Scholes a Nobel Prize for the application of
Brownian motion in finance, which is called Black–Scholes–Merton model.

**Definition**: Brownian motion characterized by
the following:

- At time 0, the initial value is zero
- It has stationary non-overlapping independent increment
- The increment is Normally distributed
- It has almost sure continuous path. There is no jump.

To be more precise, we will use mathematical notation of the definition above. Let be a Brownian motion (or Wiener Process), then

- = 0
- Suppose we have, then the incrementand are two independent random variables. A stochastic process is called stationary if .
- Suppose we have, then the increment a normal random variable with zero mean and variance.
- The path of over time is a continuous function with probability = 1.

Notice that the definition of Brownian motion only include the distribution of the increment but it does not include the distribution of the Brownian motion itself. What is the distribution of Brownian motion? Let us the first and third characteristics of Brownian motion above to define what the distribution of Brownian motion is. We set, we have the increment to be . That means Brownian motion is normally distributed with zero mean and variance. The distribution of would be the same as. The variance is proportional to the length of interval [s, t]. However, in general.

We can also write Brownian motion as

=

To simulate Brownian motion with time step 1, we use uniformly distributed random number to the inverse of cumulative normal distribution to produce.

Brownian motion also has a scaling property such that has the same distribution as.

Note
that Brownian motion is only continuous path but not smooth. To get the length
of the path, we use integral. However, because the path is not smooth, it is
not differentiable. Therefore, we cannot use ordinary integration (i.e. Riemann
integration). Fortunately, there is a way to do integration for non-smooth path
and that integration is called *Ito integration*.

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**
Preferable reference for this tutorial is
**

Teknomo, Kardi. (2017) Stochastic Process Tutorial .

http://people.revoledu.com/kardi/tutorial/StochasticProcess/