On Boundedness, Continuity, and Hyers--Ulam Stability of Interval-Valued Stochastic Processes in a Probabilistic Framework

Abstract

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.