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A Random Riemann–Liouville Integral Operator

Author

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  • Jorge Sanchez-Ortiz

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
    These authors contributed equally to this work.)

  • Omar U. Lopez-Cresencio

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
    These authors contributed equally to this work.)

  • Martin P. Arciga-Alejandre

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

  • Francisco J. Ariza-Hernandez

    (Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
    These authors contributed equally to this work.)

Abstract

In this work, we propose a definition of the random fractional Riemann–Liouville integral operator, where the order of integration is given by a random variable. Within the framework of random operator theory, we study this integral with a random kernel and establish results on the measurability of the random Riemann–Liouville integral operator, which we show to be a random endomorphism of L 1 [ a , b ] . Additionally, we derive the semigroup property for these operators as a probabilistic version of the constant-order Riemann–Liouville integral. To illustrate the behavior of this operator, we present two examples involving different random variables acting on specific functions. The sample trajectories and estimated probability density functions of the resulting random integrals are then explored via Monte Carlo simulation.

Suggested Citation

  • Jorge Sanchez-Ortiz & Omar U. Lopez-Cresencio & Martin P. Arciga-Alejandre & Francisco J. Ariza-Hernandez, 2025. "A Random Riemann–Liouville Integral Operator," Mathematics, MDPI, vol. 13(15), pages 1-11, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2524-:d:1718444
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    References listed on IDEAS

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    1. Alvarez-Ramirez, Jose & Alvarez, Jesus & Rodriguez, Eduardo & Fernandez-Anaya, Guillermo, 2008. "Time-varying Hurst exponent for US stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6159-6169.
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