n-dimensional correlated Brownian motions

Generate a matrix of increments from a multidimensional Brownian motion with a given vector of means and a Variance-Covariance matrix.

Functions generate a matrix of increments from a multidimensional Brownian motion with a given vector of means and a Variance-Covariance matrix.

Problem

Often when using for example multifactor models, the model requires correlated sources of noise. A popular choice is to use a multidimensional Brownian motion.

Solution

The proposed algorithm uses the fact that increments of a BM are normally distributed as well as the fact that assuming n independent BM's whose increments are generated from a standard normal distribution (denoted , a derived process has its increments distributed as where mu is the vector of desired means and L is the square root of the desired Variance-Covariance matrix (denoted E).

Inputs

• Vector of means for each BM `mu`
• Variance-Covariance matrix whose diagonal elements describe the volatility of each BM and the off-diagonal elements describe the covariance `E`
• Number of samples needed `sampleSize`

Output

• Matrix of samples where each column represents a BM and each row a new increment

Getting started

The user is interested in generating samples from 2 Brownian motions with a correlation of 0.8. Additionally, the first BM has a mean of 1 and a variance of 1.5. The second BM has a mean of 0 and a variance of 2. The user is interested in 480 samples.

Example of use

mu = [1; 0];

VarCovar = [1.5, 0.8; 0.8, 2];

sampleSize = 480;

CorBrownian(mu, VarCovar, sampleSize)