Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

Sanchez-Ortiz, Jorge; Lopez-Cresencio, Omar U.; Ariza-Hernandez, Francisco J.; Arciga-Alejandre, Martin P.

doi:10.3390/math9131479

Open AccessArticle

Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

by

Jorge Sanchez-Ortiz

^†

,

Omar U. Lopez-Cresencio

^†,

Francisco J. Ariza-Hernandez

^*,†

and

Martin P. Arciga-Alejandre

^†

Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N Cd, Universitaria, Chilpancingo, Guerrero C.P. 39087, Mexico

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2021, 9(13), 1479; https://doi.org/10.3390/math9131479

Submission received: 12 May 2021 / Revised: 8 June 2021 / Accepted: 22 June 2021 / Published: 24 June 2021

(This article belongs to the Special Issue Stochastic Processes and Random Fields)

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Abstract

In this note, we define an operator on a space of Itô processes, which we call Caputo-Itô derivative, then we considerer a Cauchy problem for a stochastic fractional differential equation with this derivative. We demonstrate the existence and uniqueness by a contraction mapping argument and some examples are given.

Keywords: brownian motion; Caputo-Itô derivative; Itô process; existence; uniqueness

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MDPI and ACS Style

Sanchez-Ortiz, J.; Lopez-Cresencio, O.U.; Ariza-Hernandez, F.J.; Arciga-Alejandre, M.P. Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative. Mathematics 2021, 9, 1479. https://doi.org/10.3390/math9131479

AMA Style

Sanchez-Ortiz J, Lopez-Cresencio OU, Ariza-Hernandez FJ, Arciga-Alejandre MP. Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative. Mathematics. 2021; 9(13):1479. https://doi.org/10.3390/math9131479

Chicago/Turabian Style

Sanchez-Ortiz, Jorge, Omar U. Lopez-Cresencio, Francisco J. Ariza-Hernandez, and Martin P. Arciga-Alejandre. 2021. "Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative" Mathematics 9, no. 13: 1479. https://doi.org/10.3390/math9131479

APA Style

Sanchez-Ortiz, J., Lopez-Cresencio, O. U., Ariza-Hernandez, F. J., & Arciga-Alejandre, M. P. (2021). Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative. Mathematics, 9(13), 1479. https://doi.org/10.3390/math9131479

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Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

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