Explicit Ornstein-Uhlenbeck solution

The solution ofisIt has variance $(1-e^{-2\lambda t})/(2\lambda )$. Starting from $N(0,(2\lambda {)}^{-1})$ makes it stationary with covariance $e^{-\lambda |t-s|}/(2\lambda )$.

For $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$ with ${\sigma }_0=0$, the explicit stochastic integral is ${\sigma }_t=\sqrt{\kappa }{\int }_0^t{e}^{-(t-u)/\tau }d{W}_u$. The Itô isometry on the overlap of the two integration intervals gives the displayed covariance. The second exponential records the zero initial condition; removing it gives the stationary Ornstein-Uhlenbeck process covariance.