Functions generates a matrix of increments from a multidimensional Brownian motion with a given vector of means and a Variance-Covariance matrix.
Offten when using for example multifactor models, the model requires correlated sources of noise. A popular choice is to use a multidimensional Brownian motion.
The proposed algorithm uses the fact that increments of a BM are normaly distributed as well as the fact that assuming n independent BM's whose increments are generated from a standard normal distribution (denoted N(0,1)), a derived proces Y = mu + L*z has its increments distributed as N(mu, E) where mu is the vector of desired means and L is the square root of the desired Variance-Covariance matrix (denoted E).
- Vecor 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
- Matrix of samples where each column represents a BM and each row a new increment
The user is interested in generating samples from 2 Brownian motions with a correlation of 0.8. Additionaly, 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.
mu = [1; 0];
VarCovar = [1.5, 0.8; 0.8, 2];
sampleSize = 480;
CorBrownian(mu, VarCovar, sampleSize)