t, x) \nu(\omega, {\operatorname{d}\hspace{-0.003cm}}t, {\operatorname{d}\hspace{-0.003cm}}x),\end{aligned}$$ is ${{\mathbf{P}}}$-indistinguishable of the $({\mathfrak{F}}, {{\mathbf{P}}})$-characteristics of $X^{\tau_n} \in \mathcal{S}(\mathbb{H}, {\mathfrak{F}}, {{\mathbf{P}}})$ for all $n \in \mathbb{N}$, then we call the triplet $(B(h), C, \nu)$ the *$({\mathfrak{F}}, {{\mathbf{P}}}, A)$-characteristics* of $X$.

The most important consistency observations concerning the $A$-characteristics, such as *existence* and *uniqueness*, and some additional notations, such as the *continuous local martingale part on $A$*, are given in Appendix \[facts HSVSM SI\].

Semimartingale Problems on Stochastic Invervals {#Semimartingale Problems on Predictable Stochastic Invervals}
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We denote the Borel $\sigma$-field of $\mathbb{H}$ by $\mathcal{H}$. In order to deal with possibly explosive processes we introduce the notion of a *grave*. For a Polish space $E$, i.e. in particular for $\mathbb{H}$ and for $\mathcal{N}(\mathbb{H}, \mathbb{H})$,[^5] let $\Delta$ be a point outside of $E$ and denote $E_\Delta = E\cup \{\Delta\}$. We define $\mathbb{D}^{E_\Delta}$ as the space of all functions $\alpha : \mathbb{R}^+ \to E_\Delta$, such that $\alpha(s) = \Delta,$ for all $s \in [\xi(\alpha), \infty)$, where $\xi(\alpha) := \inf(t \geq 0 : \alpha(t) = \Delta)$, which are [càdlàg ]{}on $(0, \xi(\alpha))$. We assume that we are given the following:

1.  an ${\mathfrak{F}}$-adapted process $X$ with paths in $\mathbb{D}^{\mathbb{H}_\Delta}$, which we call *candidate process*,

2.  a triplet $(B(h), 