An Anticipative Linear Filtering Equation

11 Pages Posted: 13 Sep 2010

See all articles by Knut K. Aase

Knut K. Aase

Norwegian School of Economics (NHH) - Department of Business and Management Science

Terje Bjuland

Norwegian School of Economics (NHH) - Department of Finance

Bernt Oksendal

University of Oslo - Department of Mathematics; Norwegian School of Economics (NHH) - Department of Finance

Date Written: August 31, 2010

Abstract

In the classical Kalman-Bucy filter and in the subsequent literature so far, it has been assumed that the initial value of the signal process is independent of both the noise of the signal and of the noise of the observations.The purpose of this paper is to prove a filtering equation for a linear system where the (normally distributed) initial value X0 of the signal process Xt has a given correlation function with the noise (Brownian motion Bt) of the observation process Zt. This situation is of interest in applications to insider trading in finance. We prove a Riccati type equation for the mean square error S(t):= E[(Xt - ^Xt)**2]; 0 <= t <= T; where ^Xt is the filtered estimate for Xt. Moreover, we establish a stochastic differential equation for ^Xt based on S(t). Our method is based on an enlargement of filtration technique, which allows us to put the anticipative linear filter problem into the context of a non-anticipative two-dimensional linear filter problem with a correlation between the signal noise and the observation noise.

Keywords: Anticipative linear filter equation, enlargement of filtration, insider trading

Suggested Citation

Aase, Knut K. and Bjuland, Terje and Oksendal, Bernt, An Anticipative Linear Filtering Equation (August 31, 2010). NHH Dept. of Finance & Management Science Discussion Paper No. 2010/8. Available at SSRN: https://ssrn.com/abstract=1676127 or http://dx.doi.org/10.2139/ssrn.1676127

Knut K. Aase

Norwegian School of Economics (NHH) - Department of Business and Management Science ( email )

Helleveien 30
Bergen, NO-5045
Norway

Terje Bjuland

Norwegian School of Economics (NHH) - Department of Finance ( email )

Helleveien 30
N-5045 Bergen
Norway

Bernt Oksendal (Contact Author)

University of Oslo - Department of Mathematics ( email )

P.O. Box 1053
Blindern, N-0162, Os
Norway
+47-2285 5913 (Phone)
+47-2285 4349 (Fax)

Norwegian School of Economics (NHH) - Department of Finance

Helleveien 30
N-5045 Bergen
Norway

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