Bounded Girsanov density for exit of an unstable linear diffusion ($Z_t=\exp \left(\frac{1}{2}{X}_{t\wedge T}^2-\frac{1}{2}(t\wedge T)-\frac{1}{2}{\int }_0^{t\wedge T}{X}_s^{2}ds\right)$)

Under a reference measure where $X$ is Brownian motion started at zero, let $T$ be its exit time from $(-1,1)$. The displayed density is the stochastic exponential with integrand $X{1}_{[0,T]}$, by the Itô formula. It is bounded by $e^{1/2}$, hence defines a uniformly integrable density process with terminal density $Z_T$. Under that measure change, $X_t-{\int }_0^{t\wedge T}{X}_{s}ds$ is Brownian motion. At exit, $Z_T\ge e^{1/2-T}$, implying $\mathbb{P}(T\le t)\ge e^{1/2-t}\tilde{\mathbb{P}}(T\le t)$. This is a terminal density on the whole probability space, not merely an unspecified finite-horizon change of measure.