Parametric Inequalities for s-Convex Stochastic Processes via Caputo Fractional Derivatives

This paper establishes a general parametric integral identity involving (n+1)-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities for differentiable s-convex stochastic processes. Remarkably, by assigning specific values to the underlying parameter, we have ensured our results specialize to well-known numerical integration inequalities, including those of midpoint, trapezium, Simpson, and Bullen types, in the stochastic fractional context. The findings not only enrich the theory of stochastic fractional calculus but also provide a flexible analytical apparatus for uncertainty quantification in fractional dynamical systems.