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The Semi-Markov property of Continuous Time Random Walks (CTRWs) and theirlimit processes is utilized, and the probability distributions of the bivariateMarkov process $(X(t),V(t))$ are calculated: $X(t)$ is a CTRW limit and $V(t)$a process tracking the age, i.e. the time since the last jump. For a given CTRWlimit process $X(t)$, a sequence of discrete CTRWs in discrete time is givenwhich converges to $X(t)$ (weakly in the Skorokhod topology). Master equationsfor the discrete CTRWs are implemented numerically, thus approximating thedistribution of $X(t)$. A consequence of the derived algorithm is that anydistribution of initial age can be assumed as an initial condition for the CTRWlimit dynamics. Four examples with different temporal scaling are discussed:subdiffusion, tempered subdiffusion, the fractal mobile/immobile model and thetempered fractal mobile/immobile model.

A Semi-Markov Algorithm for Continuous Time Random Walk Limit Distributions