  We consider stochastic and open quantum systems with a finite number of
states, where a stochastic transition between two specific states is monitored
by a detector. The long-time counting statistics of the observed realizations
of the transition, parametrized by cumulants, is the only available information
about the system. We present an analytical method for reconstructing generators
of the time evolution of the system compatible with the observations. The
practicality of the reconstruction method is demonstrated by the examples of a
laser-driven atom and the kinetics of enzyme-catalyzed reactions. Moreover, we
propose cumulant-based criteria for testing the non-classicality and
non-Markovianity of the time evolution, and lower bounds for the system
dimension. Our analytical results rely on the close connection between the
cumulants of the counting statistics and the characteristic polynomial of the
generator, which takes the role of the cumulant generating function.
