Filtering problems with exponential criteria for general Gaussian signals

TL;DR

This paper derives explicit solutions for discrete-time filtering problems with exponential criteria for general Gaussian signals using a conditional Cameron-Martin approach, providing recursive formulas and exploring special cases.

Contribution

It introduces a novel method based on a conditional Cameron-Martin formula to solve exponential filtering problems for Gaussian signals, including recursive solutions and special case analysis.

Findings

Explicit recursive formulas for filtering with exponential criteria

Derivation of a conditional Cameron-Martin type formula

Analysis of particular cases for the general solution

Abstract

The explicit solution of the discrete time filtering problems with exponential criteria for a general Gaussian signal is obtained through an approach based on a conditional Cameron-Martin type formula. This key formula is derived for conditional expectations of exponentials of some quadratic forms of Gaussian sequences. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem which is used in the proof. Closed form recursions of Volterra type for these ingredients are provided. Particular cases for which the results can be further elaborated are investigated.