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.

中文翻译：

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离散时间约束 Ito 过程建模的随机量化方法

物理学中的随机量化被认为是提供 Ito 过程概率分布的路径积分表示。已经表明，如果 Ito 过程仅限于朗之万方程，则随机量化可能涉及潜在项。在本文中，为了将随机量化应用于工程问题，我们提出了一种将潜在项合并到一般 Ito 过程的随机量化中的新方法。该方法表明加权分布产生了离散时间路径积分的潜在项，并保留了路径积分作为概率分布的作用，而不对漂移项进行任何假设。路径积分的随机波动的二阶近似给出了表示随机过程的期望值和协方差矩阵的时间演化的差分方程。差分方程通过识别具有惩罚或势垒函数的势函数，显式地推导出扩展卡尔曼滤波器和受约束 Ito 过程的模型。约束随机系统的数值模拟表明，势项可以将非线性动力学约束在势函数的最小值或递减方向上。差分方程通过识别具有惩罚或势垒函数的势函数，显式地推导出扩展卡尔曼滤波器和受约束 Ito 过程的模型。约束随机系统的数值模拟表明，势项可以将非线性动力学约束在势函数的最小值或递减方向上。差分方程通过识别具有惩罚或势垒函数的势函数，显式地推导出扩展卡尔曼滤波器和受约束 Ito 过程的模型。约束随机系统的数值模拟表明，势项可以将非线性动力学约束在势函数的最小值或递减方向上。