Lecture Notes for Simulation

27 April 2005 - Markov Chains

Alternatively (and equivalently), Pij(m)(t) can be computed via the Chapman-Komogorov equations

Pij(n)(t) = sum(k = 0 to M, Pik(r)(t)Pkj(n-r)(t))

for all 0 < r < m via the following reasoning:

Pij(n)(t)=P(Xt+n = j | Xt = i)
=sum(k in S, P(Xt+n = j and Xt+r = k | Xt = i))
=sum(k in S, P(Xt+n = j | Xt+r = k and Xt = i)P(Xt+r = k | Xt = i))
=sum(k in S, Pkj(n-r)(t)Pik(r)(t)

This page last modified on 2 March 2005.