  We consider the evolution of a quantum simple harmonic oscillator in a
general Gaussian state under simultaneous time-continuous weak position and
momentum measurements. We deduce the stochastic evolution equations for
position and momentum expectation values and the covariance matrix elements
from the system's characteristic function. By generalizing the
Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 2015) to
this continuous variable system, we construct its stochastic Hamiltonian and
action. Action extremization gives us the equations for the most-likely readout
paths and quantum trajectories. For steady states of the covariance matrix
elements, the analytical solutions for these most-likely paths are obtained.
Using the CDJ formalism we calculate final state probability densities exactly
starting from any initial state. We also demonstrate the agreement between the
optimal path solutions and the averages of simulated clustered stochastic
trajectories. Our results provide insights into the time dependence of the
mechanical energy of the system during the measurement process, motivating
their importance for quantum measurement engine/refrigerator experiments.
