= Stationary drift saturation
{title2=$dZ=\sqrt{2\lambda}\,dW+g\,dt,\quad \mathbb EZ_t^2=1,\quad\mathbb Eg_t^2\le\lambda^2$}
For an adapted <stationary process> with the displayed bounds, the <Itô formula> and stationarity give $\int_0^t(\mathbb E[Z_sg_s]+\lambda)ds=0$. The <Cauchy-Schwarz inequality> makes each integrand nonnegative, so equality holds almost everywhere. Then $\mathbb E(g_s+\lambda Z_s)^2\le0$, proving $g=-\lambda Z$ for almost every time and outcome. Thus the process solves the <Ornstein-Uhlenbeck stochastic differential equation> and its initial value, rather than stationarity alone, determines its coupling with W.
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