Formal calculus for real‐valued fractional Brownian motions prospects in systems science

Abstract

PurposeTo define the main elements of a formal calculus which deals with fractional Brownian motion (fBm), and to examine its prospects of applications in systems science.Design/methodology/approachThe approach is based on a generalization of the Maruyama's notation. The key is the new Taylor's series of fractional order f(x+h)=Eα(hαDα)f(x), where Eα( · ) is the Mittag‐Leffler function.FindingsAs illustrative applications of this formal calculus in systems science, one considers the linear quadratic Gaussian problem with fractal noises, the analysis of the equilibrium position of a system disturbed by a local fractal time, and a model of growing which involves fractal noises. And then, one examines what happens when one applies the maximum entropy principle to systems involving fBms (or shortly fractals).Research limitations/implicationsThe framework of this paper is applied mathematics and engineering mathematics, and the results so obtained allow the practical analysis of stochastic dynamics subject to fractional noises.Practical implicationsThe direct prospect of application of this approach is the analysis of some stock markets dynamics and some biological systems.Originality/valueThe fractional Taylor's series is new and thus so are all its implications.