We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.

中文翻译：

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一维扩散过程的最优过滤问题

我们找到了一种方法，该方法可以将一维扩散过程的非线性过滤问题的求解简化为具有恒定扩散系数的线性抛物方程的求解，该线性抛物方程的剩余系数是随机的，并且取决于可观察过程的轨迹。该方法包括将初始滤波问题简化为具有恒等扩散矩阵的更简单问题，然后将抛物线Itô方程的滤波密度减小至上述抛物线方程。另外，两个问题的滤波密度通过足够简单的公式连接。