 Ornstein-Uhlenbeck multiplicative amplification

Let $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$, ${\sigma }_0=0$, and $\dot{B}=\sigma B$, $B_0>0$. The logarithm $\log (B_t/B_0)={\int }_0^t{\sigma }_{s}ds$ is a centered Gaussian random variable of variance $V(t)=\kappa {\tau }^2[t-2\tau (1-e^{-t/\tau })+\tau (1-e^{-2t/\tau })/2]$. Squaring the stochastic kernel $\sqrt{\kappa }\tau [1-e^{-(t-u)/\tau }]$ and integrating proves this formula. The exponential moment of a Gaussian linear functional gives $\mathbb{E}{B}_t^n=B_0^n{e}^{n^{2}V(t)/2}$ and the displayed long-time growth exponent. Positive moment growth with zero mean strain reflects the breadth of the log-normal distribution; it does not imply positive growth of the mean logarithm.