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 $P$, let $X$ be standard Brownian motion started at zero and let $T$ be its first exit from $(-1,1)$. The Itô formula makesthe stochastic exponential with integrand $X{1}_{[0,T]}$. Since $Z_t\le e^{1/2}$, it is a uniformly integrable martingale and defines $dQ=Z_{T}dP$ on the whole original sigma-algebra. The Girsanov theorem makes $X_t-{\int }_0^{t\wedge T}{X}_{s}ds$ a Brownian motion under $Q$.

At exit, $|X_T|=1$ and ${\int }_0^T{X}_s^{2}ds\le T$, so $e^{1/2-T}\le Z_T\le e^{1/2}$. On $\{T\le t\}$, integrate these likelihood bounds in $Q(T\le t)={\mathbb{E}}_P[Z_T{1}_{\{T\le t\}}]$ to obtainA lower bound on $Z_T=dQ/dP$ gives an upper bound on $P$ in terms of $Q$; reversing that direction gives an invalid comparison.