Ornstein-Uhlenbeck Process (OU Process)

The Ornstein-Uhlenbeck (OU) process is a type of stochastic process used to model the evolution of a variable over time with a tendency to revert toward a long-term mean

$d{X}_t=\theta (\mu -X_t)dt+\sigma d{W}_t$

• $X_t$ is the variable at time $t$

• $\mu$ is the long-term mean

• $\theta$ i the rate of reversion

• $\sigma$ is the volatility

• $d{W}_t$ is a Wiener process increment

• $\theta$ rate of reversion It determines how fast the process “pulls” back to the mean.

• Volatility coefficient Determines the magnitude of the random fluctuations (the “noise”). Controls how “rough” or “smooth” the trajectory looks.