The Brownian motion is one of the most famous and important stochastic process.

Let’s start with a bit of history.

Robert Brown discovered the Brownian motion unintentionally while observing movements of grains of pollen through a microscope in 1827. He noticed that pollen seeds suspended in water moved in an irregular zigzag and random manner.

Louis Bachelier was the first person to model it. He used it to model the dynamic of stock prices and value stock options in his thesis in 1900. Five years later, Albert Einstein modeled the trajectory of atoms subject to shocks with random motion and obtained a Gaussian density.

A stochastic process W(t) is a Brownian motion, also called Wiener process if:

1) W(0) is equal to zero

2) It has a continuous path

3) It has independent and stationary increments

4) the increments follow a Gaussian distribution. W(t+u) – W(t) follows a gaussian distribution with zero mean and variance u.

The concept of martingale is essential in financial mathematics for the determination of the prices of financial products. A stochastic process is a martingale if the conditional expectation of S(t+s) given all the information known at t is equal to S(t). Below are several examples of martingale involving Wiener processes:

– W(t) is a martingale

– W(t)^2 – t is also a martingale

– exp(lambda x W(t) – 1/2 x lambda^2 x t) is a martingale

An important result is that the quadratic variation of a Brownian motion is equal to t with a probability 1. It is the cornerstone of Ito and stochastic calculus. We will not go through the demonstration here, we illustrate it with simulations.

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