  This paper proposes a governing equation for stock market indexes that
accounts for 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 by evaluating the
Hurst exponent of the multifractal time series, its power spectrum, and its
autocorrelation.
