%% Simulating the SDE using Euler - Maruyama Method .
for i=0:1000
    lambda = 0.5;
    q=1;
    x=1;
    mu = 0;
    t=0;
    dt =1/100;
    steps = 1/ dt;
    X = x;
    X1 =1;
    T = t;

    for k=1: steps
        fx = - lambda *x;
        dv = mu + sqrt (q*dt) * randn ;


        x = x + fx*dt + dv;
        t = t + dt;
        X = [X x];
        T = [T t];


    end


    plot(T,X,'black'), hold on
    means=mean(X);
    stds=std(X);
end


    plot(T,1-T.*means,'red')
    plot(T,1+T*3.*stds,'blue')
    plot(T,1-T*3.*stds,'blue')
    plot(T,1+(T*fx*dt),'green')