%Simulation d'un mouvement brownien multidimensionnel 
%dont les composantes sont corrélées
%à l,aide de la décomposition de Choleski

clear all; clc;

n_traj = 20000;               %Number of paths
n_per =10;                    %Number of time periods
Delta_t = 0.1;                %length of a time step
T=n_per*Delta_t;              %Maturity date

d = 3;                       %dimension of the Brownian motion
%correlation matrix
Gamma = [1 0.2 0.8; 0.2 1 0.5; 0.8 0.5 1]
choleski_G = chol(Gamma)     %Choleski decomposition

%Initialisation of the matrices
W = zeros(n_per+1,d);
test = zeros(n_per+1,d);

%Simulation de chaque trajectoire du mouvement brownien multidimensionnel
for j=1:n_traj
   Z = normrnd(0,1,n_per,d);       %mxd matrix of N(O,1) r.v.
   for i = 1 : n_per               %Multidimensionnal Brownian motion
      W(i+1,:) = W(i,:) + sqrt(Delta_t)*Z(i,:)*choleski_G;
   end
   test(j,:) = W(n_per+1,:);
   end

%calcul des matrices de covariance
covarianceW = cov(test)
correlationW = covarianceW / T