论文标题
Kolmogorov方程与非线性滤波方程相关的概率度量措施：粘度方法
Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach
论文作者
论文摘要
我们研究了与非线性滤波的Kushner-Stratonovich方程相关的概率措施空间。我们证明了粘度意义上的存在和独特性，尤其是我们提供了比较定理。在随机过滤的背景下，考虑满足随机微分方程的度量值的过程是很自然的。在文献中，解决此问题的经典方法是假设这些测量值的过程允许密度。我们的方法是不同的，我们直接与措施合作。因此，我们研究的向后kolmogorov方程是抛物面型抛物面类型的二阶部分微分方程，并具有紧凑的支持的概率测量空间。在文献中，对于此类问题的粘度解决方案，尤其是唯一性是一个非常具有挑战性的问题。在这里，我们找到了粘度解决方案，然后证明它是通过比较定理唯一的。
We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.
