In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also very convenient mathematically, since it allows further useful analysis to be done, such as transition rates in Markov processes.

I recently read a paper ( https://arxiv.org/abs/1811.07401 ) where a stochastic process is described, with probabilities being discontinuous at an infinite (!!) number of points. According to the paper, it shows the problematic nature of a certain class of algorithms with which the process is associated.

Are there any continuous-time stochastic processes with discontinuous transition probabilities or is it fundamentally incompatible with mathematical/physical reality?? I would like an answer for stochastic processes describing real phenomena/systems, not for theoretically/artificially constructed processes.

quantumMarkov process. I suspect the same holds for the classical version. If you are interested I can post an answer. $\endgroup$ – lcv Dec 18 '18 at 7:28