Exercise Set 2, Question 4

sigma=[0.1:0.02:0.6];
prob=zeros(size(sigma)); 

Tmax=10;
N=200;
h=Tmax/N;
t=0:h:Tmax;
M=10000;  
Z=randn(M,N);

for k=1:length(sigma)

    X=[0.5*ones(M,1),zeros(M,N)];
    sig=sigma(k);

    for j=1:M
        for i=1:N
            if (abs(X(j,i))<10)
                X(j,i+1)=X(j,i)+h*X(j,i)*(1-X(j,i))+sig*sqrt(h)*Z(j,i);
            else
                X(j,i+1)=10;
            end
        end
    end
    
    prob(k)=sum(X(:,N+1)==10)/M;
    
end

figure (1)
plot(sigma,prob);

Running gave the following plot

We expect the probability of divergence to increase with sigma - higher fluctuations gives more chance of divergence.

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