Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck (OU) process is a stochastic differential equation describing a mean-reverting random process:

$$dV=-kVdt+dW,k>0$$

where $dW$ is an increment of the Wiener process. The drift term $-kVdt$ pulls the process back towards zero.

Numerical Simulation

Applying the Euler-Maruyama scheme:

$$V(t+dt)=(1-kdt)V(t)+\sqrt{dt}\cdot Z,Z\sim N(0,1)$$

Equilibrium Distribution

As $t\to \infty$, the OU process approaches a stationary (equilibrium) distribution:

$$V(\infty )\sim N\left(0,\frac{1}{2k}\right)$$

The process is mean-reverting: the mean is always zero, and the variance is controlled by the parameter $k$. Larger $k$ means stronger mean reversion and a tighter equilibrium distribution.

The OU process is the basis of the Langevin equation model for particle velocity in a thermal bath, and its first time to reach a threshold is described by First-passage time.

Stochastic differential equation | Wiener process | Euler-Maruyama scheme | Langevin equation | First-passage time