A Pathwise Solution of the Equations of Nonlinear Filtering
A Pathwise Solution of the Equations of Nonlinear Filtering
复制标题
非线性滤波方程组的路径解
DOI：
10.1137/1127017
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发表时间：
1982
影响因子：
0.6
通讯作者：
Mark H. A. Davis
中科院分区：
文献类型：
--
作者：
Mark H. A. Davis
最近，许多作者--例如，Doss [4]，Sussman [12]--证明了具有光滑系数和标量”噪声”的随机微分方程的解可以以不直接涉及随机积分的方式表示为噪声样本路径的泛函。本文的目的是得到非线性滤波的一般方程的类似表示，该方程可按Kunita [7]的方法视为测度值随机微分方程。相关的过滤理论在下面的2中概述; Liptser和Shiryaev [9]可以参考富士的故事。当信号过程伊萨有限状态马尔可夫链时，所需的表示由下面的（2.10）给出，并且是[4]结果的直接应用;它已经由Rozovskii和Shiryaev [10]（他们涵盖了可数状态空间的情况）和Clark [2]从稍微不同的观点获得。在定理2中，在总结中提到的一般公式在某些条件下被证明是适当的，并且在定理3中，使用Stroock [11]的结果，当信号过程（xi）由L 6vy生成器控制时，这些条件被满足。当（x，）是一个扩散过程时，不需要求助于”鞅问题”结果”Girsanov测度变换技术的简单应用就足够了，这给出了由观测过程样本路径参数化的函数空间积分形式的条件期望，这一形式已由Clark [2]得到，也可在[9]的定理8.7的证明中找到。我们的结论与过滤的扩散与边界条件的一些意见。
Recently, a number of authors--forexample, Doss [4], Sussman [12]--have shown that the solution of a stochastic differential equation with smooth coefficients and a scalar" noise" can be expressed as a functional of the noise sample path in a way that does not directly involve stochastic integration. The purpose of this paper isto obtain a similar representation for the general equation of nonlinear filtering which, following Kunita [7], can be regarded as a measure-valued stochastic differential equation. The relevant filtering theory is outlined in 2 below; Liptser and Shiryaev [9] can be consulted for the fuji story. When the signal process isa finite-stateMarkov chain, the desired representation is given by (2.10) below and is a direct application of the results of [4]; it was already obtained, from somewhat different points of view, by Rozovskii and Shiryaev [10](who cover the countable state-space case) and by Clark [2]. In Theorem 2 the general formula referred to in the Summary is shown to be the appropriate one under certain conditions, and in 3 it is shown that these conditions are satisfied when the signal process (x,) is governed by a L6vy generator, using results ofStroock [11]. When (x,) is a diffusion process, it is not necessary to appeal to" martingale problem" results" a simple application of the Girsanov measure transformation technique suffices, and this gives the conditional expectation in the form of a function-space integral parameterized by the observation process sample path, a form which was already obtained by Clark [2] and which can also be found in the proof of Theorem 8.7 of [9]. We conclude with some remarks on the filtering of diffusions with boundary conditions.