ON THE LINEAR-EXPONENTIAL FILTERING PROBLEM FOR GENERAL GAUSSIAN PROCESSES

• Published in: SIAM journal on control and optimization Vol. 47; no. 6; pp. 2886 - 2911
• Main Authors: KLEPTSYNA, M. L, LE BRETON, A, VIOT, M
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2008
• Subjects: Applied sciences; Brownian motion; Computer science; control theory; systems; Control theory. Systems; Exact sciences and technology; Laplace transforms; Mathematics; Modelling and identification; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Statistics; Statistics Theory; Stochastic processes; Itô equation; 60G44; Closed form equation; Stochastic process; Stochastic integral; Inference mechanisms; Covariance; Gaussian signal; Primary; Linear filtering; exponential criteria; System identification; Volterra equation; 62M20; Secondary; Riccati-Volterra equation; Continuous process; Gaussian process; Filter; risk-sensitive filtering; Optimal filtering; Riccati equation; Risk management; Quadratic functional; 60G15; filtering error; optimal filtering
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/070705908

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