%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 dt.

randn('state',100)
lambda = 2                        % problem parameters
mu = 1; 
Xzero = 1;
T = 1; 
N = 2^8; 
dt = 1/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

Xem = zeros(1,N);                 % preallocate for efficiency

Xem(1) = Xzero + dt*lambda*Xzero + mu*Xzero*dW(1);

for j=2:N
   Xem(j) = Xem(j-1) + dt*lambda*Xem(j-1) + mu*Xem(j-1)*dW(j);
end

plot([0:dt:T],[Xzero,Xem],'b--*'), hold off
xlabel('t','FontSize',12)
ylabel('x','FontSize',16,'Rotation',0,'HorizontalAlignment','right')

emerr = abs(Xem(end)-Xtrue(end))