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

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.

Space-time fractional porous media equation: Application on modeling of S&P500 price return.

We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of "fractionalization" are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized q-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that q-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)1063-651X10.1103/PhysRevE.99.062313]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.