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

Abstract

Purpose

To 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/approach

The 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)=E_α(h^αD^α)f(x), where E_α( · ) is the Mittag‐Leffler function.

Findings

As 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/implications

The 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 implications

The direct prospect of application of this approach is the analysis of some stock markets dynamics and some biological systems.

Originality/value

The fractional Taylor's series is new and thus so are all its implications.

Citation

Jumarie, G. (2006), "Formal calculus for real‐valued fractional Brownian motions prospects in systems science", Kybernetes, Vol. 35 No. 9, pp. 1393-1413. https://doi.org/10.1108/03684920610662430