Ornstein-Uhlenbeck stochastic differential equation
= Ornstein-Uhlenbeck stochastic differential equation
{c}
{title2=$dY_t=\sigma\,dW_t-\lambda Y_t\,dt$}
The linear <stochastic differential equation> has the explicit solution $Y_t=e^{-\lambda t}Y_0+\sigma\int_0^te^{-\lambda(t-s)}dW_s$. Its integrating factor is $e^{\lambda t}$. For $\lambda>0$, an independent initial normal variable of variance $\sigma^2/(2\lambda)$ makes its zero-mean <Gaussian process> stationary.