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Define the stochastic integralIt is adapted to ${\mathcal{G}}_t$, starts at zero and has continuous paths. Integrals over disjoint clock intervals are centered Gaussian and independent, and an increment from $s$ to $t$ is independent of ${\mathcal{G}}_s$. The Itô isometry gives its varianceThese are precisely the defining properties of a ${\mathcal{G}}_t$-Brownian motion; in particular its future increments are independent of ${\mathcal{G}}_0={\mathcal{F}}_1$.

Deterministic substitution in the stochastic integral, first for step integrands and then by the Itô isometry, gives $d{X}_s=\sqrt{2\lambda }{e}^{\lambda s}d{W}_s$. HenceThe initial term is essential: a stochastic integral from time zero alone cannot equal $X$, since $X_0=B_1$. This is the unscaled part of the exponential Brownian time change to a stationary Ornstein-Uhlenbeck process.

Use the Itô product rule with the deterministic factor $e^{-\lambda t}$. Its quadratic covariation with $X$ is zero, givingMultiplication by the integrating factor $e^{\lambda t}$ solves the Ornstein-Uhlenbeck stochastic differential equation:The initial value $B_1$ is standard normal and independent of the future driving Brownian motion. Thus $Y$ is a centered Gaussian process with variance one. For $s\le t$, its independent future noise gives $\mathbb{E}(Y_s{Y}_t)=e^{-\lambda (t-s)}$. Its Gaussian finite-dimensional distributions consequently depend only on time differences, so it is a stationary process. This verifies the exponential Brownian time change to a stationary Ornstein-Uhlenbeck process directly.