statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, a generalized Wiener process (named after

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

) 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 "poi ...

random walk with drift and

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

jumps at every point in time. Formally: :

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

where a and b are

deterministic Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...

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 is the opposite of endogeneity or endogeny, the fact of being influenced from within a system. Economics In an economic model, an ...

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

draw from a

standard normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^ ...

at each instant.

See also