I am not familiar with the deep mathematical intricacies of advanced no-arbitrage theory, an extremely technical subject. However, from reading literature reviews, I suspect this is an historical legacy of the research path that led to the most general versions of no-arbitrage theory.
If you consider dividend-paying assets whose dividends are not continuously reinvested, then you actually need to model the dynamics of your asset using a jump process to represent the out-coming (dividend, coupon) cash flow: indeed, if $S_t$ is your stock price process, then when a dividend of amount $D$ is paid at $t^-$, then the price of the stock should immediately jump downards:
$$S_{t^+}=S_{t^-}-D$$
It turns out introducing jump processes does seem to compound considerably the mathematical difficulty of formalising no-arbitrage theory. The seminal paper proving the most general version of the no-arbitrage theorem is "A general version of the fundamental theorem of asset pricing" (Delbaen and Schachermayer, 1993). In page 2, they write (my emphasis):
We believe that the main theorem (Theorem 1.1 below) of this paper contributes to both theories mathematics as well as economics. In economic terms the theorem contains essentially two messages. First that it is possible to characterise the existence of an equivalent martingale measure for a general class of processes in terms of the concept of no free lunch with vanishing risk, a concept to be defined below. In this notion the aspect of vanishing risk bears economic relevance. The second message is that - in a general setting - there is no way to avoid general stochastic integration theory. If the model builder accepts the possibility that the price process has jumps at all possible times, he needs a sophisticated integration theory, going beyond the theory for "simple
integrands". In particular the integral of unbounded predictable processes of general nature has to be used. [...]
Further, in pages 4 and 5 they write (my emphasis):
To relate our work to earlier results, let us summarise the present state of the art. The case when the time set is finite is completely settled in Dalang et al. (1989) and the use of simple or even elementary integrands is no restriction at all (see Schachermayer (1992), Kabanov and Kramkov (1993) and Rogers (1993) for elementary proofs). For the case of discrete but infinite time sets, the problem is solved in Schachermayer (1993). The case of continuous and bounded processes in continuous time, is solved in Delbaen (1992). In these two cases the theorems are stated in terms of simple integrands and limits of
sequences and by using the concept of no free lunch with bounded risk. We shall review these issues in Sect. 6.
In the general case, i.e. a time set of the form $[0,\infty[$ or $[0, 1]$ and with a possibility of random jumps, the situation is much more delicate. [...]
