represents a stationary Ornstein–Uhlenbeck process with long-term mean μ, volatility , and mean reversion speed θ.
represents an Ornstein–Uhlenbeck process with initial condition x0.
- OrnsteinUhlenbeckProcess is a continuous-time and continuous-state random process.
- OrnsteinUhlenbeckProcess is also known as Vasicek model.
- The state of an Ornstein–Uhlenbeck process satisfies an Ito differential equation , where follows a standard WienerProcess.
- The initial value for OrnsteinUhlenbeckProcess[μ,σ,θ] is random and follows NormalDistribution[μ,σ/].
- OrnsteinUhlenbeckProcess[μ,σ,θ] value at time t follows NormalDistribution[μ,σ/].
- OrnsteinUhlenbeckProcess[μ,σ,θ,x0] value at time t follows NormalDistribution[ μ+(x0-μ)exp(-θ t),].
- OrnsteinUhlenbeckProcess allows μ and x0 to be any real numbers and σ and θ to be any positive real numbers.
- OrnsteinUhlenbeckProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Examples (3)
Basic Uses (7)
Process Slice Properties (5)
CentralMoment and its generating function:
FactorialMoment has no closed form for symbolic order:
Cumulant and its generating function:
Generalizations & Extensions (1)
Properties & Relations (9)
OrnsteinUhlenbeckProcess starting at a random value is weakly stationary:
Ornstein–Uhlenbeck process has a well-defined StationaryDistribution:
An Ornstein–Uhlenbeck process with a fixed initial condition is a special ItoProcess:
As well as StratonovichProcess:
Ornstein–Uhlenbeck process at integer times behaves as a first-order ARProcess:
Create moment equations to find parameters for an ARProcess:
Create an ARProcess: