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A Stochastic Fractional Calculus with Applications to Variational Principles

by Houssine Zine †,‡ and Delfim F. M. Torres *,‡
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
*
Author to whom correspondence should be addressed.
This research is part of first author’s Ph.D. project, which is carried out at the University of Aveiro under the Doctoral Program in Applied Mathematics of Universities of Minho, Aveiro, and Porto (MAP).
These authors contributed equally to this work.
Fractal Fract 2020, 4(3), 38; https://doi.org/10.3390/fractalfract4030038
Received: 19 May 2020 / Revised: 14 July 2020 / Accepted: 30 July 2020 / Published: 1 August 2020
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. View Full-Text
Keywords: fractional derivatives and integrals; stochastic processes; calculus of variations fractional derivatives and integrals; stochastic processes; calculus of variations
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Zine, H.; Torres, D.F.M. A Stochastic Fractional Calculus with Applications to Variational Principles. Fractal Fract 2020, 4, 38.

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