Ornstein–Uhlenbeck process

E48273

Gaussian process Markov process continuous-time process mean-reverting process stochastic process

The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.

Observed surface forms (2)

Surface form	As subject	As object
Hull–White interest rate model	0	1
Vasicek interest rate model	0	1

Statements (49)

Predicate	Object
instanceOf	Gaussian process → Markov process → continuous-time process → mean-reverting process → stochastic process →
alsoKnownAs	OU process →
appliedIn	commodity price modeling → neuroscience membrane potential models → term structure modeling → thermal fluctuations → velocity of a Brownian particle →
describes	fluctuations around equilibrium → mean-reverting random motion →
field	probability theory → quantitative finance → statistical physics → stochastic calculus →
generalizationOf	discrete-time AR(1) process in continuous time →
governedBy	stochastic differential equation →
hasAutocorrelationFunction	exponentially decaying autocorrelation →
hasCovarianceStructure	depends only on time difference in stationary regime →
hasDriftForm	linear drift toward long-term mean →
hasMeanFunction	exponential reversion to long-term mean →
hasNoiseTerm	additive Brownian motion →
hasParameter	long-term mean → speed of mean reversion → volatility parameter →
hasProperty	Gaussian transition densities → Markov processes → surface form: Markov property continuous sample paths → ergodic under suitable parameters → mean reversion → stationary increments only in the limit of infinite time → time-homogeneous →
hasStationaryDistribution	normal distribution →
introducedIn	1930s →
language	mathematics →
namedAfter	George Eugene Uhlenbeck → Leonard Ornstein →
relatedTo	Brownian motion → Ornstein–Uhlenbeck process self-linksurface differs → surface form: Hull–White interest rate model Ornstein–Uhlenbeck process self-linksurface differs → surface form: Vasicek interest rate model
solutionOf	linear stochastic differential equation with constant coefficients →
specialCaseOf	Markov processes → surface form: Gaussian Markov process Langevin dynamics → surface form: Langevin equation
usedIn	Brownian motion with friction → Langevin dynamics → modeling interest rates → modeling volatility →

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Euler–Maruyama method → relatedTo → Ornstein–Uhlenbeck process →

Fokker–Planck equation → relatedTo → Ornstein–Uhlenbeck process →

Itô processes → relatedTo → Ornstein–Uhlenbeck process →

subject surface form: Itô process

Ornstein–Uhlenbeck process → relatedTo → Ornstein–Uhlenbeck process self-linksurface differs →

this entity surface form: Vasicek interest rate model

Ornstein–Uhlenbeck process → relatedTo → Ornstein–Uhlenbeck process self-linksurface differs →

this entity surface form: Hull–White interest rate model