T = 100000; t = 0;
lambda = 2; mu = 0.25;

X = 0; jump_times = [0]; jump_chain = [X];

while t <= T
    t = t - log(rand) / (lambda + X*mu);
    if t > T
        break;
    else
        jump_times = [jump_times t];
        
        if rand < lambda / (lambda + X*mu)
            X = X + 1;
        else
            X = X - 1;
        end
        jump_chain = [jump_chain X];
    end
end