  In this paper, we develop the elements of the theory of algorithmic
randomness in continuous-time Markov chains (CTMCs). Our main contribution is a
rigorous, useful notion of what it means for an $\textit{ individual trajectory
}$ of a CTMC to be ${ \textit random }$. CTMCs have discrete state spaces and
operate in continuous time. This, together with the fact that trajectories may
or may not halt, presents challenges not encountered in more conventional
developments of algorithmic randomness.
  Although we formulate algorithmic randomness in the general context of CTMCs,
we are primarily interested in the $\textit{ computational }$ power of
stochastic chemical reaction networks, which are special cases of CTMCs. This
leads us to embrace situations in which the long-term behavior of a network
depends essentially on its initial state and hence to eschew assumptions that
are frequently made in Markov chain theory to avoid such dependencies.
  After defining the randomness of trajectories in terms of a new kind of
martingale (algorithmic betting strategy), we prove equivalent
characterizations in terms of constructive measure theory and Kolmogorov
complexity. As a preliminary application, we prove that, in any stochastic
chemical reaction network, $\textit{ every }$ random trajectory with bounded
molecular counts has the $\textit{ non-Zeno property }$ that infinitely many
reactions do not occur in any finite interval of time.
