Bilinear Stochastic Systems with Long Range Dependence in Continuous Time

Abstract

Application of a long range dependent model includes several fields of science and economics as geophysics, hydrology, turbulence, weather and so on. Recently it has been successfully used for modeling network traffic data (see [WTLW]). The basic stochastic process of this kind is the fractional Brownian motion defined in [MvN]. The fractional Brownian motion is given as a particular fractional operator on the standard Brownian motion. The linear or Gaussian parametric models of long range dependent phenomena are both the linear stochastic differential equations with fractional Brownian motion input and the fractional operator on the solution of a linear stochastic differential equation (see [C]). Actually these two types of processes are equivalent. Because most of the observations are not Gaussian there is a need nonlinear modeling of long range dependence. One possibility is to get rid of Gaussianity is the bilinear model started by Subba Rao [SR] in the discrete time case. The easy way to get a long range non-Gaussian process is to apply the fractional operator to the solution of the bilinear SDE. It is more painful to consider a bilinear SDE with fractional Brownian motion input.KeywordsFractional OperatorFractional Brownian MotionStochastic IntegrationRange DependenceWhite Noise InputThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.