Variable order porous media equations: Application on modeling the S&P500 and Bitcoin price return

Abstract

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.