A unified multi-parameter fractional integral framework for probabilistic inequalities and moment analysis with applications to reliability and financial risk

Abdelkader Moumen; Jehad Alzabut; Oualid Zentar; Abdulhamit Kucukaslan; Mahrouz Tayeb; Mohamed Bouye; Abdelkader Moumen; Jehad Alzabut; Oualid Zentar; Abdulhamit Kucukaslan; Mahrouz Tayeb; Mohamed Bouye

doi:10.3934/math.20261040

AIMS Mathematics

2026, Volume 11, Issue 8: 25958-25983. doi: 10.3934/math.20261040

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A unified multi-parameter fractional integral framework for probabilistic inequalities and moment analysis with applications to reliability and financial risk

1.
Department of Mathematics, Faculty of Sciences, University of Ha'il, Ha'il 55425, Saudi Arabia
2.
Department of Industrial Engineering, OSTİM Technical University, 06374 Ankara, Türkiye
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
4.
Department of Computer Science, Laboratory of Research in Artificial Intelligence and Systems (LRAIS), University of Ibn Khaldoun, Tiaret 14000, Algeria
5.
Department of Mathematics, Faculty of Engineering and Natural Sciences, Ankara Yılıdırım Beyazıt University, Ankara, Türkiye
6.
Department of Mathematics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received: 09 April 2026 Revised: 10 June 2026 Accepted: 28 July 2026 Published: 20 August 2026
MSC : 26A33, 26A51, 26D10

This paper develops a unified probabilistic framework based on a generalized multi-parameter fractional integral operator. The operator incorporates an auxiliary function, a kernel, a scale parameter, and a fractional order, providing a flexible mechanism for modeling memory-dependent processes. Using this operator, we introduce generalized fractional expectation, variance, and higher-order moments for random variables and establish fundamental inequalities in this setting, including generalized Grüss-, Chebyshev-, Markov-, and Hoeffding-type (concentration) inequalities. The proposed framework unifies many existing fractional models as special cases and extends them through additional parameters and kernel structures. To illustrate applicability, we discuss models arising in reliability analysis and financial risk assessment. These examples demonstrate how generalized fractional operators can be used to quantify uncertainty, volatility, and risk in systems where memory and non-uniform weighting play an important role.
- fractional expectation,
- fractional variance,
- fractional moment,
- generalized fractional integral operator,
- integral inequality,
- random variable
Citation: Abdelkader Moumen, Jehad Alzabut, Oualid Zentar, Abdulhamit Kucukaslan, Mahrouz Tayeb, Mohamed Bouye. A unified multi-parameter fractional integral framework for probabilistic inequalities and moment analysis with applications to reliability and financial risk[J]. AIMS Mathematics, 2026, 11(8): 25958-25983. doi: 10.3934/math.20261040

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Abstract

This paper develops a unified probabilistic framework based on a generalized multi-parameter fractional integral operator. The operator incorporates an auxiliary function, a kernel, a scale parameter, and a fractional order, providing a flexible mechanism for modeling memory-dependent processes. Using this operator, we introduce generalized fractional expectation, variance, and higher-order moments for random variables and establish fundamental inequalities in this setting, including generalized Grüss-, Chebyshev-, Markov-, and Hoeffding-type (concentration) inequalities. The proposed framework unifies many existing fractional models as special cases and extends them through additional parameters and kernel structures. To illustrate applicability, we discuss models arising in reliability analysis and financial risk assessment. These examples demonstrate how generalized fractional operators can be used to quantify uncertainty, volatility, and risk in systems where memory and non-uniform weighting play an important role.

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