There is by now an extensive and well-developed theory of weak convergence for moving averages and continuous-time random walks (CTRWs) with respect to Skorokhod's M1 and J1 topologies. Here we address the fundamental question of how this translates into functional limit theorems, in the M1 or J1 topology, for stochastic integrals driven by these processes. As a key application, we provide weak approximation results for a general class of SDEs driven by time-changed L\'evy processes. Such SDEs and their associated fractional Fokker--Planck--Kolmogorov equations are central to models of anomalous diffusion in statistical physics, and our results provide a rigorous functional characterisation of these as continuum limits of the corresponding models driven by CTRWs. In regard to strictly M1 convergent moving averages and so-called correlated CTRWs, it turns out that the convergence of stochastic integrals can fail markedly. Nevertheless, we are able to identify natural classes of integrand processes for which the convergence holds. We end by showing that these results are general enough to yield functional limit theorems, in the M1 topology, for certain stochastic delay differential equations driven by moving averages.

翻译：目前，关于滑动平均和连续时间随机游走（CTRWs）在Skorokhod M1和J1拓扑下的弱收敛理论已发展得相当完善。本文探讨一个基本问题：这些收敛性如何转化为由这些过程驱动的随机积分在M1或J1拓扑下的函数极限定理。作为关键应用，我们为一类由时间变换Lévy过程驱动的随机微分方程（SDEs）提供了弱近似结果。这类SDEs及其相关的分数阶Fokker-Planck-Kolmogorov方程是统计物理学中反常扩散模型的核心，而我们的结果为这些方程作为相应CTRW驱动模型的连续极限提供了严格的函数刻画。对于严格M1收敛的滑动平均和所谓的相关CTRWs，随机积分的收敛性可能显著失效。尽管如此，我们仍能识别出使收敛性成立的自然被积过程类。最后，我们证明这些结果足够普适，能够为某些由滑动平均驱动的随机延迟微分方程在M1拓扑下建立函数极限定理。