there exists an increasing sequence $(S_n)_{n \in \mathbb{N}}$ of $\mathbb{F}$-stopping times such that $S = \lim_{n\rightarrow +\infty} S_n$ and the process $(X^{S_n},\mathbb{F})$ obtained by stopping $(X,\mathbb{F})$ in $S_n$ is a real-valued semimartingale for every $n\in\mathbb{N}$. Analogously, we introduce the notion of a *local martingale up to $S$*.\
If $(X,\mathbb{F})$ is a semimartingale up to $S$, then we can find a decomposition $$\label{eqn:semi_decomposition}
    X_t = X_0 + M_t + V_t\,, \qquad t < S, \ \mathbf{P}\text{-a.s.},$$ on $[0,S)$, where $(M,\mathbb{F})$ is a local martingale up to $S$ with $M_0 = 0$ and $(V, \mathbb{F})$ is a right-continuous process whose paths are of bounded variation on $[0,t]$ for every $t < S$ and with $V_0=0$. If $X$ is continuous on $[0,S)$, then there exists a decomposition such that $M$ and $V$ are continuous on $[0,S)$ and this decomposition is unique on $[0,S)$. For any continuous local martingale $(M, \mathbb{F})$ up to $S$ by ${    \ensuremath{\langle M \rangle}}$ we denote the continuous increasing process, which is uniquely determined on $[0,S)$, such that $(M^2 - {    \ensuremath{\langle M \rangle}}, \mathbb{F})$ is a continuous local martingale up to $S$ and ${    \ensuremath{\langle M \rangle}}_0 = 0$. For a continuous semimartingale $(X,\mathbb{F})$ up to $S$ we 