# The brownian motion with implementation (C# and R)

## Theory

Brownian motion is the presumably random moving of particles suspended in a fluid (a liquid or a gas)

resulting from
their bombardment by the fast-moving atoms or molecules in the gas or liquid.

The term "Brownian motion" can also refer to the mathematical model used to describe such random movements, which is often called a particle theory.
The main proble with this constinuous process is its implementation for making experiences. The distribution of its increment is the principal tool which people use to implement it.

If we assume that B(t) is a brownian, motion, B(t+ Δt)-B( t) ∼ N(0, Δt). Where N represent the normal distribution with mean 0 and variance Δt

## Implementation

## The One Dimensional Brownian

Firstly, we need the simulation of the gaussian standard distribution the C# code is:

Now we can implement our one dimensional brownian motion. The C# code is:

The above code may be easily set with the R-Language because the default functions we can find in it.
This is the R code:

## The Multi Dimensional Brownian with independant components

We are going to use the above function (the R one) to do this simulation Here the R code:

## The Multi Dimensional Brownian with correlated components

To get correlated brownians we have to have first independant Brownians. With the VarCoV matrix, we perform

the Cholesky decomposition, and finally we can easily get our correlated brownian with a simple matrix product.

Copyright : Moulong Armel Rodrigue, 2012

Contact: moulongarmel@yahoo.fr