Generator and jump chain

Topic: Continuous-time Markov chains · more questions on this topic
‹ A two-state chain in continuous time Why a generator row sums to zero ›

States \{0,1,2\} have rates 0\to 1 equal to 2, 1\to 0 equal to 1, 1\to 2 equal to 3, and 2\to 1 equal to 4. Write the generator and the jump-chain probabilities.

Worked answer (click to show)

Exit rates are \nu_0=2, \nu_1=1+3=4, and \nu_2=4. The generator and the non-zero jump probabilities are

Q=\begin{pmatrix}-2&2&0\\ 1&-4&3\\ 0&4&-4\end{pmatrix}, \qquad k_{01}=1,\quad k_{10}=\frac14,\quad k_{12}=\frac34,\quad k_{21}=1.