  Subharmonic response is a well known phenomena in, e.g., deterministic
nonlinear dynamical systems. We investigate the conditions under which such
subharmonic oscillations can persist for a long time in open systems with
stochastic dynamics due to thermal fluctuations. In contrast to stochastic
autonomous systems in a stationary state, for which the number of coherent
oscillations is fundamentally bounded by the number of states in the underlying
network, we demonstrate that in periodically driven systems, subharmonic
oscillations can, in principle, remain coherent forever, even in networks with
a small number of states. We also show that, {\sl inter alia}, the
thermodynamic cost rises only logarithmically with the number of coherent
oscillations in a model calculation and that the possible periods of the
persistent subharmonic response grow linearly with the number of states. We
argue that our results can be relevant for biochemical oscillations and for
stochastic models of time-crystals.
