set ${    \ensuremath{\langle X \rangle}} = {    \ensuremath{\langle M \rangle}}$, where $M$ is the continuous local martingale up to $S$ in the decomposition (\[eqn:semi\_decomposition\]) of $X$.\
We now recall some facts which are well-known for continuous semimartingales (see for example [@revuzyor], Ch. VI, §1). Their extension to semimartingales $(X,\mathbb{F})$ up to an $\mathbb{F}$-stopping time $S$ is obvious. For $(X,\mathbb{F})$ there exists the right (resp., left, symmetric) local time $L^X$ up to $S$ which is a function on $[0,S) \times \mathbb{R}$ into $[0,+\infty)$ such that for every real function $f$ which is the difference of convex functions the *generalized Itô formula* holds: $$\label{eqn:gen_ito_formula}
    f(X_t) = f(X_0) + \int_0^t f^\prime (X_s) \, {\mbox{\upshape d}}X_s + \frac{1}{2} \int_0^t L^X(t,y) \, {\mbox{\upshape d}}f^\prime(y)\,, \qquad t < S, \ \mathbf{P}\text{-a.s.}$$ where $f^\prime$ denotes the left (resp., right, symmetric) derivative of $f$. To indicate explicitly which local time we consider, we write $L_+^X$ (resp., $L_-^X$, $\hat{L}^X$) for the right (resp., left, symmetric) local time of $X$. If a formula or statement holds for every type of local time we just use the symbol $L^X$. Note that we can choose $L_+^X$ (resp., $L_-^X$) to be increasing and continuous in $t < S$ and right (resp. left) continuous 