F. Levernhe et al., Markovian diffusive representation of 1/f(alpha) noises and application tofractional stochastic differential models, IEEE SIGNAL, 49(2), 2001, pp. 414-423
This paper is devoted to linear stochastic differential systems with fracti
onal noise (or fractional Brownian motion) input. On the basis of a conveni
ent Markovian description of such noises, elaborated from a diffusive repre
sentation of fractional integrators previously introduced in a deterministi
c context, the fractional differential system is equivalently transformed i
nto a standard (but infinite-dimensional) one, with white-noise input. Fini
te-dimensional approximations mag. easily be obtained from classical discre
tization schemes.
With this equivalent representation, the correlation function of processes
described by linear fractional stochastic differential systems may be expre
ssed 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 stocha
stic systems.