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Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 53104, 26 pages
http://dx.doi.org/10.1155/JAMSA/2006/53104

Linear filtering of systems with memory and application to finance

1Department of Mathematics, Faculty of Science, Hokkaido Universit, Sapporo 060-0810, Japan
2Division of Mathematical Sciences for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan
3School of Mathematical Sciences, Queensland University of Technology, G.P.O. Box 2434, Brisbane, Queensland 4001, Australia

Received 29 July 2004; Revised 25 March 2005; Accepted 5 April 2005

Copyright © 2006 A. Inoue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the linear filtering problem for systems driven by continuous Gaussian processes V(1) and V(2) with memory described by two parameters. The processes V(j) have the virtue that they possess stationary increments and simple semimartingale representations simultaneously. They allow for straightforward parameter estimations. After giving the semimartingale representations of V(j) by innovation theory, we derive Kalman-Bucy-type filtering equations for the systems. We apply the result to the optimal portfolio problem for an investor with partial observations. We illustrate the tractability of the filtering algorithm by numerical implementations.