  We study the dynamical properties of a driven-dissipative Bose-Hubbard model
in the strongly interacting regime through a quantum trajectory approach with a
cluster-Gutzwiller Ansatz for the wave function. This allows us to take
classical and quantum correlations into account. By studying the dynamical
hysteresis surface that arises by sweeping through the coherent driving
strength we show that the phase diagram for this system is in qualitative
correspondence with the Gutzwiller mean-field result. However, quantitative
differences are present and the inclusion of classical and quantum correlations
causes a significant shift of the critical parameters. Additionally, we show
that approximation techniques relying on a unimodal distribution such as the
mean field and $1/z$ expansion drastically underestimate the particle number
fluctuations. Finally, we show that a proposed mapping of the
driven-dissipative many-body Bose-Hubbard model onto a single
driven-dissipative Kerr model is not accurate for parameters in the hysteresis
regime.
