Zero-start Ornstein-Uhlenbeck covariance ($\mathbb{E}[{\sigma }_t{\sigma }_s]=\frac{\kappa \tau }{2}(e^{-|t-s|/\tau }-e^{-(t+s)/\tau })$)

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.