Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions

Stochastic dynamics of fractional order are usually modeled as non-random differential equation driven by fractional Brownian motion. Here we propose rather to use a non-random fractional dynamics driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x) where Eα(·)Eα(·) denotes the Mittag–Leffler function, and Dxα is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. The equivalence of the two models is clarified, and one shows how to switch from one of them to the other one. Two illustrative examples are displayed, which are the stochastic differential equations defining fractional coloured noises on the other hand, and fractional exponential growth on the other hand.