What is the definition for the process $X_-$?

Question

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.

Answer 1

$X_-$ is the process of left limits (assuming they exist): $(X_-)_t=X_{t-}:=\lim_{s\to t, s<t} X_s$, for $t>0$. The process $X_-$ is left continuous.

If $X$ is cadlag and adapted, then $X_-$ is predictable. And the sequence of stopping times $(T_n)$ defined by $$ T_n:=\inf\{t:|X_{t-}|>n\} $$ bears witness to the fact that $X_-$ is locally bounded, because $X_-$ stopped at time $T_n$ is bounded in magnitude by $n$ and $\lim_n T_n=+\infty$.