Longitudinal studies of secular events are frequently conceptualized as consisting of a trend component fitted by a polynomial, a cyclical component often fitted by an analysis based on autocorrelations or on a Fourier series, and a random component (stochastic drift) to be removed.
Autocorrelation analysis helps to identify the correct phase of the fitted model while the successive differencing transforms the stochastic drift component into white noise.
Time series variables in economics and finance — for example, stock prices, gross domestic product, etc.
In this case the stochastic term is stationary and hence there is no stochastic drift, though the time series itself may drift with no fixed long-run mean due to the deterministic component f(t) not having a fixed long-run mean.
using a functional form coinciding with that of f, and retaining the stationary residuals.
In contrast, a unit root (difference stationary) process evolves according to where
But even in the absence of the parameter c (that is, even if c=0), this unit root process exhibits drift, and specifically stochastic drift, due to the presence of the stationary random shocks ut: a once-occurring non-zero value of u is incorporated into the same period's y, which one period later becomes the one-period-lagged value of y and hence affects the new period's y value, which itself in the next period becomes the lagged y and affects the next y value, and so forth forever.
So after the initial shock hits y, its value is incorporated forever into the mean of y, so we have stochastic drift.
In the context of monetary policy, one policy question is whether a central bank should attempt to achieve a fixed growth rate of the price level from its current level in each time period, or whether to target a return of the price level to a predetermined growth path.
In the latter case no price level drift is allowed away from the predetermined path, while in the former case any stochastic change to the price level permanently affects the expected values of the price level at each time along its future path.