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Stochastic quantization in physics has been considered to provide a pathintegral representation of a probability distribution for Ito processes. It hasbeen indicated that the stochastic quantization can involve a potential term,if the Ito process is limited to Langevin equation. In this paper, in order toapply the stochastic quantization to engineering problems, we propose a novelmethod to incorporate a potential term into stochastic quantization of thegeneral Ito process. This method indicates that weighted distribution givesrise to the potential term for the discrete-time path integral and preservesthe role of the path integral as the probability distribution, without makingany assumptions on the drift term. A second order approximation on thestochastic fluctuations for the path integral gives difference equations whichrepresent the time evolution of expectation value and covariance matrix for thestochastic processes. The difference equations explicitly derive ExtendedKalman Filter and models on the constrained Ito processes by the identificationof the potential function with a penalty or barrier function. The numericalsimulations of the constrained stochastic systems show that the potential termcan constrain the nonlinear dynamics towards a minimum or a decreasingdirection of the potential function.

Stochastic quantization approach to modeling of constrained Ito processes in discrete time