  Stochastic quantization in physics has been considered to provide a path
integral representation of a probability distribution for Ito processes. It has
been indicated that the stochastic quantization can involve a potential term,
if the Ito process is limited to Langevin equation. In this paper, in order to
apply the stochastic quantization to engineering problems, we propose a novel
method to incorporate a potential term into stochastic quantization of the
general Ito process. This method indicates that weighted distribution gives
rise to the potential term for the discrete-time path integral and preserves
the role of the path integral as the probability distribution, without making
any assumptions on the drift term. A second order approximation on the
stochastic fluctuations for the path integral gives difference equations which
represent the time evolution of expectation value and covariance matrix for the
stochastic processes. The difference equations explicitly derive Extended
Kalman Filter and models on the constrained Ito processes by the identification
of the potential function with a penalty or barrier function. The numerical
simulations of the constrained stochastic systems show that the potential term
can constrain the nonlinear dynamics towards a minimum or a decreasing
direction of the potential function.
