We consider a parametrically driven Kerr medium in which the pumping may be sinusoidally varied. It has been previously found that this system exhibits coherent tunneling between two fixed points which can be either enhanced or suppressed by altering the driving frequency and strength. We numerically investigate the dynamics when damping is included. This is done both by solving a master equation and using the quantum-trajectory method. In the latter case it is also possible to model the result of a continuous heterodyne measurement of the cavity output. The dissipation destroys the coherences which give rise to the tunneling, causing the sinusoidal oscillation of the mean to give way to a stochastic jumping between the fixed points, manifested as a random telegraph signal. In the quantum-trajectory picture we show that the coherences responsible for tunneling are an exponentially decreasing function of the signal-to-noise ratio for heterodyne measurements. However, evidence of both the bare tunneling rate and the driving modified tunneling rate are still apparent in the random telegraph signal
