Let's fix two processes, $\lambda$ and $N$.
$N$ is defined as a point process (with some fancy conditions that are not important here. It shall be right continuity as processes are usually. On the other hand, I define $\lambda$ in the following way:
$$ \lambda (t \mid \mathcal F_{t^-} ) = \int_0^t \mu( t - s ) d N_s $$
where $\mu$ is a continuous function. I am baffled because I noticed that
If one knows that $N$ has jumps $(t_1, \cdots, t_n)$ on $[0,T]$, and that the process can be written as:
$$N(t) = \sum_{ \{ k : t_k \leq t \} } 1 $$
then:
$$ \int_{[0,T]} \mu(s) d N_s =\sum_{ \{ k : t_k \leq T \} }\mu(t_k) $$
$1_A$ is the indicator/characteristic function over the set $A$.
however, $\lambda$ should be left continuous. It seems to me that by definition, the term $\sum_{ \{ k : t_k \leq T \} }\mu(t_k) $ is right continuous. It would become left continuous if inside the sum, the condition on the $k$'s was strictly lower than the bound.
What am I missing please ?
My guess for now is that it depends on the notation of $dN_s$. Depending on its definition, one has different meaning of the stochastic integral, and thus either left or right continuity.