Explicit Ornstein-Uhlenbeck solution

For deterministic $X_0=x$ and real $\lambda$, the Ornstein-Uhlenbeck stochastic differential equation $d{X}_t=-\lambda X_{t}dt+d{B}_t$ has solutionMultiply by the integrating factor $e^{\lambda t}$ and apply the Itô product rule to obtain it. The stochastic integral is centered Gaussian and its variance, by the Itô isometry, isThus the apparent zero-denominator case is ordinary Brownian motion, and negative $\lambda$ gives growing rather than mean-reverting fluctuations.

For $\lambda >0$, take an independent initial value $X_0\sim N(0,(2\lambda {)}^{-1})$. Adding its variance contribution gives the time-independent variance $(2\lambda {)}^{-1}$ and covariance function $e^{-\lambda |t-s|}/(2\lambda )$, making the process stationary. A deterministic zero start instead has the transient variance in the display. These are different initial ensembles, so their variances should not be interchanged.

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.