I often see for a stochastic process $X$ an new process denoted by $X_-$. I just know that this is left continuous, but I do not know the exact definition. Furthermore, if I have a RCLL process $X$ and looking at $X_-$ what can I say? I this process then continuous? The reason why I ask is the following:
For a RCLL process (and adapted) $K$ there is a suitable sequence of partitions, with mesh size converging to $0$ as $n\to \infty$ such that
$\int_0^t K_{s_-} dX_s = \lim_{n\to\infty}\sum K_{t_i}(X_{t_{i+1}\wedge t}-X_{t_i\wedge t})$
where $X$ is for example a continuous semimartingale. I can prove this for a bounded $K$. However to localizing I need that $K$ must be continuous. Then I can use the fact that every adapted continuous process is locally bounded. Thank you for your help.