Is expm accurate to compute transition probabilities for a continuous-time Markov chain?

Question

Yufei Cao 2021-7-7

回答： Yazan 2021-7-7

I have the following Q-matrix for a continuous time Markov chain and use expm to compute transition probabilities.

syms t positive
Q = [-2 1 1;
     1 -1 0;
     2  1 -3];
P(t) = expm(Q*t);
double(P(2))

Matlab gives

ans =
3797    0.4908    0.1295
3704    0.5092    0.1205
3794    0.4908    0.1298
    

My question is that I think ans(2,3) should be zero, becasue Q(2,3)=0. But Matlab shows that ans(2,3)=0.1205.

Thanks in advantage.

Answer 1

Yazan 2021-7-7

expm(sig) computes the matrix exponential of sig according to:

[V,D] = eig(sig)

expm(sig) = V*diag(exp(diag(D)))/V

For your example:

Q = [-2 1 1;
     1 -1 0;
     2  1 -3];
P = expm(Q*2);
[V,D] = eig(Q*2);  
P2 = V*diag(exp(diag(D)))/V;
% maximum difference
max(P2(:) - P(:))