Before revision:  Both of them are non-stationary time series. This means that their statistical distribution change over time . Consequently a detrended process is made to neutralize the non-stationary effects. The detrended process is obtained by decomposing the financial time series into a deterministic trend and random fluctuations. We present an alternative method on detrending time series based on the classical moving average (MA) models, where Kurtosis is used to determine the windows size. Then, the dentrending fluctuation analysis (DFA) is use to show that the detrended part is stationary .
After revision:  Both of them are non-stationary effects. This is a linear Fokker-Planck equation (FPE) that describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito's lemma, this FPE is associated with a stochastic differential equation (SDE) that models the time evolution of the price return in a fashion different from the classical Black-Scholes equation. Both FPE and SDE equations account for a deterministic part or trend, and a stationary, stochastic part as a q-Gaussian noise. The model is validated using the S\&P500 index's data. After removing the trend from the index, we show that the detrended part is stationary .