In this paper, we investigate interval-valued stochastic processes within the framework of pseudo-order relations and analyze their fundamental analytical properties. We give a probabilistically meaningful definition of boundedness for interval-valued processes and use it to establish a single, consolidated local-to-global boundedness theorem, together with a corrected treatment of continuity in probability and of Hyers–Ulam stability for interval-valued stochastic mappings. Every construction that relies on interval subtraction, interval absolute value, or the ordering of negative intervals is given an explicit. The proposed approach provides a consistent structure for handling uncertainty arising from both randomness and interval-valued data, and we illustrate it with a toy financial example in which a return is simultaneously random and only known up to a measurement interval. We show precisely how the developed results relate to, and in the component-wise pseudo-order case reduce to, existing real-valued results, and we discuss open problems for the genuinely set-valued (inclusion-order) setting.

In this work, we establish several new variants of the Hermite–Hadamard integral inequality, together with various product-type extensions, by employing the Prabhakar fractional integral operator with three-parameter Mittag-Leffler kernels for mappings defined on harmonic sets. This unified operator generalizes several classical fractional operators, including Riemann–Liouville, Erdélyi–Kober, and Weyl, which are recovered as special cases. The results obtained herein constitute natural generalizations and significant improvements of existing results developed using classical integral operators. To demonstrate the validity and effectiveness of the obtained results, we present nontrivial numerical examples supported by graphical illustrations and tabulated comparisons for different choices of the fractional parameters. Furthermore, as an application of our main results, we derive explicit error estimates for the trapezoidal rule on harmonic intervals.