%BPATH3  Function along a Brownian path

randn('state',100)                                % set the state of randn
T = 1; N = 500; dt = T/N; t = [dt:dt:1];

M = 1000;                                         % M paths simultaneously
dW = sqrt(dt)*randn(M,N);                         % increments
W = cumsum(dW,2);                                 % cumulative sum
U = exp(repmat(t,[M 1]) + 0.5*W);
Umean = mean(U);
plot([0,t],[1,Umean],'b-'), hold on               % plot mean over M paths
plot([0,t],[ones(5,1),U(1:5,:)],'r--'), hold off  % plot 5 individual paths
xlabel('t','FontSize',16)
ylabel('U(t)','FontSize',16,'Rotation',0,'HorizontalAlignment','right')
legend('mean of 1000 paths','5 individual paths',2)

averr = norm((Umean - exp(9*t/8)),'inf')          % sample error