Optimal nonlinear filtering for independent increment processes--Part I

Abstract

A new formulation of the multidimensional optimal nonlinear filtering problem is presented in this two-part paper. This formulation permits generalization and unification of some well-known recent results on optimal nonlinear filtering theory. [1]-[7] Specifically, the problem investigated is that of determining the conditional probability density function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{y(\tau); t_{0} \leq \tau \leq t\}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> is the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional state vector of a non-linear system perturbed by an independent increment noise process, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -dimensional measurement vector which is a nonlinear function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and contains an additive independent increment noise process. The results are obtained through use of characteristic functions and the theory of independent increment processes. The foundation for the treatment of general independent increment noise processes is given in Part I, but the final results in Part I are restricted to Gaussian independent increment noise processes. The extension to general independent increment noise processes is considered in Part II. It is shown in Part I that the results for the linear-Gaussian case can be obtained in two different ways, one of which cannot be used for the general case. Some important properties of general independent increment processes and a special property of Gaussian independent increment processes are discussed.