= Exponential Brownian time change to a stationary Ornstein-Uhlenbeck process
{title2=$Y_t=e^{-\lambda t}B_{e^{2\lambda t}}$}
For $\lambda>0$, set $\mathcal G_t=\mathcal F_{e^{2\lambda t}}$ and $W_t=(2\lambda)^{-1/2}\int_1^{e^{2\lambda t}}u^{-1/2}dB_u$. This is a <Brownian motion> in the new <filtration>; its independent Gaussian increments have variance equal to elapsed time. The <Itô product rule> gives $dY=\sqrt{2\lambda}\,dW-\lambda Y\,dt$. Its initial value is $B_1$, independent of future W increments, and its <covariance function> is $e^{-\lambda|t-s|}$. Consequently it is a stationary <Ornstein-Uhlenbeck process>. The quadratic variation of the unscaled time change is $e^{2\lambda t}-1$, not $e^{2\lambda t}$.
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