times, c.f. Proposition \[main prop uni 1\] and \[main prop uni 2\]. We use these results to formulate a boundedness condition of local type that also can be verified in cases where the conditions of Novikov- and Kazamaki-type are too strict, c.f. Section \[MSE\]. In Section \[Martingality of Generalized Stochastic Exponentials\] we investigate stochastic exponentials as explicit examples.

To illustrate these claims in a non-Markovian and infinite-dimensional setting let $$W^* := \sup_{s < \cdot} W_s\ \textup{ and }\ N := W^* \cdot W,$$ where $W$ is an (infinite-dimensional) Brownian motion. In the one-dimensinal case, we have ${{\mathbf{E}}}[\exp\{(W^*)^2/2 \cdot I_t\}] \geq {{\mathbf{E}}}[\exp\{W^2/2 \cdot I_t\}] = \infty$, for $t$ large enough, where $I_t = t$ denotes the identity process. This implies that Novikov’s condition is violated. We will see in Section \[A Case Study - Martingality in Terms of SDEs driven by Hilbert-Space-Valued Brownian Motion\] that $Z$ is in fact a true martingale. Thirdly, we show that under appropriate mild assumptions existence of a solution to the modified SMP implies martingality even if the original SMP has no global solution, c.f. Corollary \[explosion cond coro\]. This can be interpreted as a type of explosion condition as given by [@MU(2012)], but now in 