  We develop a statistical model of microscopic stochastic deviation from
classical mechanics based on a stochastic processes with a transition
probability that is assumed to be given by an exponential distribution of
infinitesimal stationary action. We apply the statistical model to
stochastically modify a classical mechanical model for the measurement of
physical quantities reproducing the prediction of quantum mechanics. The
system+apparatus always have a definite configuration all the time as in
classical mechanics, fluctuating randomly following a continuous trajectory. On
the other hand, the wave function and quantum mechanical Hermitian operator
corresponding to the physical quantity arise formally as artificial
mathematical constructs. During a single measurement, the wave function of the
whole system+apparatus evolves according to a Schr\"odinger equation and the
configuration of the apparatus acts as the pointer of the measurement so that
there is no wave function collapse. We will also show that while the result of
each single measurement event does not reveal the actual value of the physical
quantity prior to measurement, its average in an ensemble of identical
measurement is equal to the average of the actual value of the physical
quantity prior to measurement over the distribution of the configuration of the
system.
