A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Abstract

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish.

Keywords

• fractional calculus
• inequalities
• neutral stochastic differential equations
• averaging principle
• Caputo–Katugampola derivatives