The question has been tackled in different settings most prominently by integrability conditions of Novikov- and Kazamaki-type, c.f. [@J79; @KS(2002b); @kazamaki77; @LM; @Novikov73; @protter; @RufNK]. A series of papers further explored the one-dimensional diffusion setting that thanks to the work [@MU(2012)] is particularly well understood. Their main result relates martingality, i.e. the martingale property, of a generalized stochastic exponential driven by a solution $(X, W)$ to an SDE of the type $$\begin{aligned}
\label{SDE1}
{\operatorname{d}\hspace{-0.003cm}}X_t = b(X_t){\operatorname{d}\hspace{-0.003cm}}t + a(X_t){\operatorname{d}\hspace{-0.003cm}}W_t\end{aligned}$$ to exit times of a solution to a modified SDE $$\begin{aligned}
\label{SDE2}
{\operatorname{d}\hspace{-0.003cm}}Y_t = \widetilde{b}(Y_t) {\operatorname{d}\hspace{-0.003cm}}t + a(Y_t){\operatorname{d}\hspace{-0.003cm}}B_t. $$ In particular, the result distinguishes at which end of the state space $(l, r)$ the modified solution exits. A remarkable observation is that there are examples where the generalized stochastic exponential is a martingale while both the solution of the driving and the modified SDE exit their state space. In the case where it is the real line, this means that both SDEs may be explosive.

Already the topology of multidimensional state spaces does not allow a direct generalization of this subtle observation to higher dimensions. In finite-dimensional diffusion settings [@HR; @Sin] connect the martingality of a non-explosive stochastic exponential driven by a 