Linear Filtering With Fractional Noise

Filtering is an application of stochastic differential equations, which through the stoch- astic differential equations to complete a best estimate and forecast problems. Such prob- lems widely exist in the electronic technology, space science; control engineering and other science and technology departments. In the history the first consider is the Wiener filter and later is Kalman-Bucy filter which was proposed in the 20th century to the 1960s.In general, the assumption of the filter is that driven noise is the standard white noise, whose increments feature is independence, which means that past history variables and variables from the past to the present and the future evolution of the way the forecast is not relevant. It is the standard Brownian motion. But in many practical applications, variable increments are interrelated, which means that the variable is self-similarity and long relevant from the time and space. The model established in this Brownian motion can not accurately portray the reality of the development of variables. In order to solve these problems, we must find a random process whose characterization is self-similar and long relevance. So many people interested in building a stochastic model of the fractional Brownian motion. In this paper I discuss a more general filter driven by the fractional white noise based on a large of studies on the property of the fractional Brownian motion and filtering driven by the fractional white noise. I introduce the property of the fractional Brownian motion , the stochastic model driven by the fractional Brownian motion, discuss linear filtering problems which given by the stochastic differential equations on the fractional Brownian motion ,and derive an explicit expression for the filtering problem using a theorem on normal correlation and Gromwell inequality.