title: The Euler-Maruyama uses timestep Dt multiple of a step size dt
body: <p>I wonder if you could help me with this?
   why the Euler-Maruyama uses timestep Dt multiple of a step size   of the increment dt for the Brownian path ?</p>
  
  <p><a href="http://personal.strath.ac.uk/d.j.higham/em.m" rel="nofollow">http://personal.strath.ac.uk/d.j.higham/em.m</a></p>
  
  <pre><code>%EM  Euler-Maruyama method on linear SDE
  %
  % SDE is  dX = lambda*X dt + mu*X dW,   X(0) = Xzero,
  %      where lambda = 2, mu = 1 and Xzero = 1.
  %
  % Discretized Brownian path over [0,1] has dt = 2^(-8).
  % Euler-Maruyama uses timestep R*dt.
  
  randn('state',100)
  lambda = 2; mu = 1; Xzero = 1;    % problem parameters
  T = 1; N = 2^8; dt = T/N;         
  dW = sqrt(dt)*randn(1,N);         % Brownian increments
  W = cumsum(dW);                   % discretized Brownian path 
  
  Xtrue = Xzero*exp((lambda-0.5*mu^2)*([dt:dt:T])+mu*W); 
  plot([0:dt:T],[Xzero,Xtrue],'m-'), hold on
  
  R = 4; Dt = R*dt; L = N/R;        % L EM steps of size Dt = R*dt
  Xem = zeros(1,L);                 % preallocate for efficiency
  Xtemp = Xzero;
  for j = 1:L
     Winc = sum(dW(R*(j-1)+1:R*j)); 
     Xtemp = Xtemp + Dt*lambda*Xtemp + mu*Xtemp*Winc;
     Xem(j) = Xtemp;
  end
  
  plot([0:Dt:T],[Xzero,Xem],'r--*'), hold off
  xlabel('t','FontSize',12)
  ylabel('X','FontSize',16,'Rotation',0,'HorizontalAlignment','right')
  
  emerr = abs(Xem(end)-Xtrue(end))
  </code></pre>