Stochastic differential equations with fractional Brownian motion input. (English) Zbl 0771.60043

Jumarie, Guy

Int. J. Syst. Sci. 24, No. 6, 1113-1131 (1993).

Summary: Kolmogorov-Levy-Mandelbrot \((t-s)^{2a}\)-fractional Brownian motion (FBM) appears to be quite relevant for modelling long range memory stochastic systems, and the problem of defining stochastic differential equations subject to such a noise is considered. The Liouville fractional derivative and the self-similarity property of FBM are recalled and then, via detailed calculation, the main statistical characteristic of FBM are derived. First-order stochastic differential equations with FBM are considered via path integrals and a corresponding mean squares approach to nonlinear filtering is described. Lastly, a new modelling via stochastic differential equations of fractional order is suggested.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
93E03	Stochastic systems in control theory (general)

Keywords:

fractional Brownian motion input; stochastic differential equations; self-similarity property; path integrals; nonlinear filtering

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