  Stochastic difference equations and a stochastic partial differential
equation (SPDE) are simultaneously derived for the time-dependent neutron
angular density in a general three-dimensional medium where the neutron angular
density is a function of position, direction, energy, and time. Special cases
of the equations are given such as transport in one-dimensional plane geometry
with isotropic scattering and transport in a homogeneous medium. The stochastic
equations are derived from basic principles, i.e., from the changes that occur
in a small time interval. Stochastic difference equations of the neutron
angular density are constructed, taking into account the inherent randomness in
scatters, absorptions, and source neutrons. As the time interval decreases, the
stochastic difference equations lead to a system of Ito stochastic differential
equations (SDEs). As the energy, direction, and position intervals decrease, an
SPDE is derived for the neutron angular density. Comparisons between numerical
solutions of the stochastic difference equations and independently formulated
Monte Carlo calculations support the accuracy of the derivations.
