C, \nu)$, called a *candidate triplet*, consisting of\

    1.  an $\mathfrak{F}$-predictable process $B(h)$ with paths in $\mathbb{D}^{\mathbb{H}_\Delta}$,\

    2.  an $\mathfrak{F}$-predictable process $C$ with paths in $\mathbb{D}^{\hspace{0.04cm}\mathcal{N}(\mathbb{H}, \mathbb{H})_\Delta}$,\

    3.  an $\mathfrak{F}$-predictable random measure $\nu$ on $\mathbb{R}^+ \times \mathbb{H}$,\

3.  a probability measure $\eta$ on $(\mathbb{H}, \mathcal{H})$, which is called *initial law*.

The process $B(h)$ may depend on the truncation function $h$.

\[Definition Semimartingale Problem\] We call a probability measure ${{\mathbf{P}}}$ on $(\Omega, \mathcal{F})$ a *solution to the SMP* associated with $(\mathbb{H};\rho;\eta; X; B(h), C, \nu)$, where $\rho$ is an ${\mathfrak{F}}$-stopping time, $\eta$ is an initial law, $X$ a candidate process and $(B(h), C, \nu)$ a candidate triplet, if\

1.  $X^{\rho}$ is ${{\mathbf{P}}}$-indistinguishable from an $\mathbb{H}$-valued $(\mathfrak{F}, {{\mathbf{P}}})$-semimartingale,\

2.  the $({\mathfrak{F}}, {{\mathbf{P}}})$-characteristics of $X^{\rho}$ are ${{\mathbf{P}}}$-indistinguishable from $(B^{\rho}(h), C^{\rho}, \nu^{\rho})$,\

3.  ${{\mathbf{P}}}\circ X^{-1}_0 = \eta$.\

We call ${{\mathbf{P}}}$ a solution to the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$, where $A \in \mathcal{I}({\mathfrak{F}})$, if there exists an announcing sequence $(\tau_n)_{n \in \mathbb{N}}$ for $A$, such that for all $n \in \mathbb{N}$, ${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H};\tau_n; \eta; X; B(h), C, \nu)$.

${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H}; A; X; \eta; 