We obtain that $$\begin{aligned}
{{\mathbf{E}}}[Z_\infty] &\geq \lim_{n \to \infty} {{\mathbf{E}}}[Z_\infty {\mathbf{1}}_{\{\rho_n = \infty\}}] = \lim_{n \to \infty} {{\mathbf{E}}}[ {{\mathbf{E}}}[Z_\infty |\mathcal{F}_{\rho_n}] {\mathbf{1}}_{\{\rho_n = \infty\}}]
= \lim_{n \to \infty} {{\mathbf{Q}}}_n(\rho_n = \infty) = 1,\end{aligned}$$ thanks to $\{\rho_n = \infty\} \in \mathcal{F}_{\rho_n}$ and Doob’s stopping theorem. This finishes the proof.

Our main results introduce more structure for these conditions and thereby increase their applicability. In order to identify a promising structure, let us observe the following: If there exists a probability measure ${{\mathbf{Q}}}$ such that $$\begin{aligned}
{{\mathbf{Q}}}= {{\mathbf{Q}}}_n = Z_{\rho_n}\cdot {{\mathbf{P}}}\textup{ on } \mathcal{F}_{\rho_n},\textup{ for all } n \in \mathbb{N}, \end{aligned}$$ then boils down to $$\begin{aligned}
\label{cond q}
\lim_{n \to \infty} {{\mathbf{Q}}}(\rho_n > t) = 1,\textup{ for all } t \geq 0.\end{aligned}$$ Moreover, if holds and ${{\mathbf{P}}}$-a.s. $\rho_n \uparrow \infty$ as in Lemma \[ruf mimic\], then Lemma III.3.3 in [@JS] yields that ${{\mathbf{Q}}}\ll_\textup{loc} {{\mathbf{P}}}$ with density process $Z$. To use this observation and identify a structure of ${{\mathbf{Q}}}$, let us introduce an additional player, an $({\mathfrak{F}}, {{\mathbf{P}}})$-semimartingale $X$. It is well-known that ${{\mathbf{Q}}}\ll_\textup{loc}{{\mathbf{P}}}$ implies that $X$ is also an $({\mathfrak{F}}, {{\mathbf{Q}}})$-semimartingale and its characteristics are related to $X$ and $Z$ via Girsanov’s theorem. Therefore ${{\mathbf{Q}}}$ is a probability measure under which $X$ is 