On Existence Results Of Boundary Value Problems Of Caputo Fractional Difference Equations For Weak-Form Efficient Market Hypothesis

The weak-form efficient market hypothesis (EMH) raises profound issues in the theory of emerging phenomena. One fundamental question is whether log-returns and log-prices exhibit closely integer-valued dynamics when considering a set of fractional difference operators. To address these complex problems, we investigate the sufficient conditions for positive solutions to boundary value problems of Caputo fractional difference equations with nonlocal boundary conditions. These results have significant applications in determining positive rates and stock prices in finance. We focus on the boundary value problem with the Caputo fractional difference operator, derive the Green’s function, and analyze it to establish existence results using Krasnosel’skii fixed-point theorem. To account for data availability, we also reformulate the problem in a more flexible topological framework. We demonstrate the methodology using log-returns on Amazon stock prices and provide insights into EMH theory and its topological aspects.