fbm1d_FBMgenerator_分数布朗运动_

function [W,t]=fbm1d(H,n,T) % fast one dimensional fractional Brownian motion (FBM) generator % output is 'W_t' with t in [0,T] using 'n' equally spaced grid points; % code uses Fast Fourier Transform (FFT) for speed. % INPUT: % - Hurst parameter 'H' in [0,1] % - number of grid points 'n', where 'n' is a power of 2; % if the 'n' supplied is not a power of two, % then we set n=2^ceil(log2(n)); default is n=2^12; % - final time 'T'; default value is T=1; % OUTPUT: % - Fractional Brownian motion 'W_t' for 't'; % - time 't' at which FBM is computed; % If no output it invoked, then function plots the FBM. % Example: plot FBM with hurst parameter 0.95 on the interval [0,10] % [W,t]=fbm1d(0.95,2^12,10); plot(t,W) % Reference: % Kroese, D. P., & Botev, Z. I. (2015). Spatial Process Simulation. % In Stochastic Geometry, Spatial Statistics and Random Fields(pp. 369-404) % Springer International Publishing, DOI: 10.1007/978-3-319-10064-7_12 if (H>1)|(H<0) % Hurst parameter error check error('Hurst parameter must be between 0 and 1') end if nargin<2 n=2^12; % grid points else n=2^ceil(log2(n)); end if nargin<3 T=1; end r=nan(n+1,1);r(1) = 1;idx=1:n; r(idx+1) = 0.5*((idx+1).^(2*H) - 2*idx.^(2*H) + (idx-1).^(2*H)); r=[r; r(end-1:-1:2)]; % first rwo of circulant matrix lambda=real(fft(r))/(2*n); % eigenvalues W=fft(sqrt(lambda).*complex(randn(2*n,1),randn(2*n,1))); W = n^(-H)*cumsum(real(W(1:n+1))); % rescale W=T^H*W; t=(0:n)/n; t=t*T; % scale for final time T if nargout==0 plot(t,W); title('Fractional Brownian motion'); xlabel('time $t$','interpreter','latex') ylabel('$W_t$','interpreter','latex') end