This is somewhat related to my unaswered question here.
Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space equipped with a filtration $\mathbb{F}$; let us also introduce $T$ as a $\mathbb{F}$-stopping time. We define the jump process $H:\Omega\times\mathbb{R}_+\rightarrow\{0,1\}$ as follows:
\begin{align}H_t=1_{[T,\infty)}(t)\end{align}
I am interested in the jump size process $\Delta H$ defined as:
$$\Delta H_t:=H_t-H_{t^-}$$
This process is equal to $0$ a.e. except on $T$ where it is equal to $1$ $-$ so it is actually indistinguishable from zero. Based on the definition of a simple predictable process given in Chapter II of Protter (1990), $H$ is one such process: \begin{align}\Delta H_t=1_{(T,T]}(t)\end{align}
Is it possible to manipulate the process $\Delta H$? In particular, can we derive an integral representation for this process? Per the properties of stochastic integrals for semimartingales, I wonder whether the following holds: \begin{align}\Delta H_t=\int_0^t(1-H_{u^-})\text{d} H_u\end{align}