= Bounded Girsanov density for exit of an unstable linear diffusion
{title2=$Z_t=\exp\!\left(\tfrac12X_{t\wedge T}^2-\tfrac12(t\wedge T)-\tfrac12\int_0^{t\wedge T}X_s^2ds\right)$}
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 $X\mathbf1_{[0,T]}$, 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, $X_t-\int_0^{t\wedge T}X_sds$ is <Brownian motion>. At exit, $Z_T\ge e^{1/2-T}$, implying $\mathbb P(T\le t)\ge e^{1/2-t}\widetilde{\mathbb P}(T\le t)$. This is a terminal density on the whole probability space, not merely an unspecified finite-horizon change of measure.
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