%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))
