In one of the papers I was reading for my masters thesis I came across a theorem with no references.
Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ martingale $\tilde{X}_t$ such that $$X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$$
where $L$ is a random time (real valued measurable random variable) on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ and $\mathcal{F}^L_t$ is a progressive enlargement of the filtration $(\mathcal{F}_t)_{t \geq 0}$ with the random time $L$ ie $\mathcal{F}^L_t=\mathcal{F}_t \vee \{L \leq t\}$, Z^L_s is the right continuous version of the super-martingale $\{\mathbb{P}(L>u \vert \mathcal{F}_u) ,u \geq 0\}$ And finally $M^L_t$ is the martingale associated with "$L$-evaluations" i.e $M^L$ is the only square integrable martingale , such that for every square integrable ($\mathcal{F}_t$) martingale $(X_t)$ we have $$E[X_L]=E[X_{\infty}M^L_{\infty}]$$
Does somebody know what kind of a decomposition this is?And provide me an exact reference?