# How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

1. Let $X=(X_{t},t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space S={1,...,K}.

2. Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random variables.

$\textbf{Question}:$ How can I simulate random paths of the Markov process with a transition intensity matrix which is built with the above hazard rates $\alpha_{i,j}(t)$?

For example: S={1,2,3,4} with given $\alpha_{i,j}(t), i,j \in S$

Any idea is greatly appreciated, with best regards, Wolfgang

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Suppose you're in state $i$. For each $j$, let $X_j$ be an independent Exponential random variable with mean 1.
Now solve $\int_{0}^{T_j}\alpha_{i,j}(t)\ dt=X_j$ for each $i$. Whichever of the $T_j$'s is smallest, you jump from state $i$ to state $j$ at time $T_j$.
No sorry I don't. Something like this has been in the back of my mind for a while. You can check it though: compute the probability that there's a transition from $i$ to $j$ in the time interval $(t,t+dt)$ conditioned that there has been no transition up to time t. –  Anthony Quas May 31 '13 at 6:04