Ornstein–Uhlenbeck process

Ornstein–Uhlenbeck process is a type of stochastic process that describes the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck. The process is a stationary Gauss-Markov process, making it a continuous-time analog of the discrete-time autoregressive process of order 1, or AR(1) process. It is a solution to the Langevin equation with a linear restoring force and Gaussian white noise, representing a model for the velocity of a particle in a fluid undergoing Brownian motion.

Definition

The Ornstein–Uhlenbeck process \(U(t)\) can be defined as the solution to the stochastic differential equation (SDE):

\[dU(t) = \theta (\mu - U(t))dt + \sigma dW(t)\]

where:

• \( \theta > 0 \) is the rate of mean reversion,
• \( \mu \) is the long-term mean level,
• \( \sigma > 0 \) is the volatility,
• \( W(t) \) is a Wiener process, and
• \( t \) represents time.

Properties

The Ornstein–Uhlenbeck process exhibits several key properties:

• It is mean-reverting, meaning it tends to drift towards its long-term mean \( \mu \) over time.
• It is a stationary process, implying that its statistical properties do not change over time.
• It is a Markov process, indicating that future values of the process depend only on the current state, not on the path taken to arrive at that state.
• It has Gaussian increments, which means that the changes in the process over any two points in time are normally distributed.

Applications

The Ornstein–Uhlenbeck process has wide applications across various fields:

• In finance, it is used to model interest rates, currency exchange rates, and commodity prices.
• In physics, it models the velocity of a particle in a fluid.
• In biology, it can describe the fluctuation in populations or the spread of viruses.
• In engineering, it is applied in signal processing and control systems.

Mathematical Solution

The exact solution of the Ornstein–Uhlenbeck SDE is given by:

\[U(t) = U(0)e^{-\theta t} + \mu(1 - e^{-\theta t}) + \sigma \int_0^t e^{-\theta (t-s)} dW(s)\]

This solution shows how the process evolves over time from an initial state \(U(0)\).

See Also

• Stochastic process
• Wiener process
• Markov process
• Brownian motion
• Langevin equation