driftinterval

Drift interval is a term used in time-series analysis and stochastic modeling to denote a specific time window over which the deterministic trend, or drift, of a process is defined or estimated. It is often employed in models that allow the drift parameter to vary with time, such as piecewise-constant drift models or regimes determined by change points. In such settings the evolution of the process X_t on a time interval I = [t0, t1] is described by a stochastic differential equation or a discrete-time analog with a constant drift mu_I within that interval, while other parameters such as volatility may be held constant or allowed to vary.

In continuous time, a common formulation on an interval I is dX_t = mu_I dt + sigma dW_t for t in I, where mu_I is the drift on I, sigma is the diffusion coefficient, and W_t is a standard Brownian motion. The drift on I can be estimated by mu_I = (E[X_{t1} - X_{t0}])/(t1 - t0) or, in practice, by linear regression of increments on time within I. When the drift is not fixed but time-varying, the drift interval concept underpins methods that approximate the drift by a sequence of adjacent intervals or fit a piecewise-constant function.

Applications include finance for short-term return trends, physics for particles under varying forces, and ecology or biology for populations responding to changing environments. Challenges include choosing the interval partition, avoiding overfitting, and separating drift from stochastic fluctuations. Related concepts are drift in time series, regime-switching models, and non-stationary diffusion processes.