we consider the situation where $Z$ is an generalized stochastic exponential. In this case the previously established integrability condition boils down to a localized Novikov-type condition, which is suitable for applications also when classical Novikov-type conditions fail to hold. Fifthly, we present another application of our abstract results which starts with the explicit choice of $(\rho_n)_{n \in \mathbb{N}}$ to be exit times of compacta. Thanks to this assumption, we derive an explosion-type condition in the spirit of [@MU(2012)]. We impose the following additional assumption.

\[SA\] We assume that the probability measure ${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$. We denote the unique decomposition of $X(h)$ by $X(h) = X_0 + M(h) + B(h)$, c.f. in Appendix \[Semimartingale Characterstics\].

All following standing assumptions are only assumed to hold in the particular subsection.

### A Condition for Strict Local Martingality

We now consider the situation where the underlying filtered space allows an extension of every consistent family. This topological property is called *fullness*, c.f. Definition \[def full\] in Appendix \[Classical Path-Spaces\], and is for instance possessed by all standard path spaces. It enables us to extend the consistent family $({{\mathbf{Q}}}_t, \mathcal{F}_t)_{t \geq 0}$ 