statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a generalized Wiener process (named after

Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

) is a

continuous time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...

random walk with drift In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

and

random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...

jumps at every point in time. Formally: :

a(x,t) dt + b(x,t) \eta \sqrt

where a and b are

deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...

functions, t is a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

index for time, x is a set of

exogenous In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system. Economics In an economic model, an exogeno ...

variables that may change with time, dt is a differential in time, and η is a

draw from a standard normal distribution at each instant.

See also