I am sharing my code,which I have tried to perform as per the instruction mentioned below. Please correct it so that I could get my output.
Instruction:
Use the first m dimensions of the Sobol vector of length (m*n) to build the realisations of m uncorrelated diffusions W_{i}(T) for i=1..m, the next m for the realisations W_{i}(T/2) and so on.
Having all the values W_{i,k} for i=1..m and k=1..n, with j representing the time point (k*ΔT) with ΔT:=T/n, subtract along time to obtain increments ΔW_{i}( from k-1 to k ).
Compute a spectral decomposition of the single-step covariance matrix C over a single time step ΔT. The elements of this single step matrix are
c_{ij} := σ_i * ρ_ij * σ_j * ΔT.
The matrices S and Λ such that C = S*Λ*S^T, the spectral-split matrix A you need is given by A := S*sqrt(Λ), where sqrt(Λ) is the diagonal matrix Λ of (non-negative) eigenvalues.
Code : This is for 20paths of 200 timesteps each . Note: Wiener process covariance is min{i,j}
m=20;
soboldata= BlockRandom[ SeedRandom[ Method -> {"MKL", Method -> {"Sobol", "Dimension" -> 20}}];
RandomReal[1, {200, 20}]]
mat= Table[min[i,j],{i,1,20},{j,1,200}];
mat//matrix form
mat= C
cov=CrM[C];
vars=Table[Symbol["x" <> ToString[i]], {i, m}];
lab=Table["Ito Plot No " <> ToString[i], {i, m}];
ip=ItoProcess[{soboldata},t,cov];
path=RandomFunction[ip,{0,1,0.05},10,Method->"Milstein"]; Evaluate@Table[ListLinePlot[path["PathComponent", i], PlotRange -> All,PlotLabel->lab[[i]],ImageSize->150],{i,m}]
