Markovian diffusive representation of 1/fα noises and application to fractional stochastic differential models

This paper is devoted to linear stochastic differential systems with fractional noise (or fractional Brownian motion) input. On the basis of a convenient Markovian description of such noises, elaborated from a diffusive representation of fractional integrators previously introduced in a deterministic context, the fractional differential system is equivalently transformed into a standard (but infinite-dimensional) one, with white-noise input. Finite dimensional approximations may easily be obtained from classical discretization schemes. With this equivalent representation, the correlation function of processes described by linear fractional stochastic differential systems may be expressed from the solution of standard differential systems, which generalizes, in some way, the well-known differential Lyapunov equation, which appears when computing the covariance matrix associated with standard linear stochastic systems.