There is by now an extensive 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 an important application, we provide weak approximation results for general 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. Our results yield a rigorous functional characterisation of these as continuum limits of the underlying models driven by CTRWs. In regard to strictly M1 convergent moving averages and 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 M1 convergence holds. We show 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：关于移动平均和连续时间随机游走（CTRW）在Skorokhod M1和J1拓扑下的弱收敛性，目前已建立了较为完备的数学理论。本文旨在解决一个关键问题：如何将这些收敛性质转化为这些过程驱动的随机积分在M1或J1拓扑下的泛函极限定理。作为重要应用，我们给出了由时间变换Lévy过程驱动的一般随机微分方程（SDE）的弱逼近结果。此类SDE及其对应的分数阶Fokker-Planck-Kolmogorov方程是统计物理中反常扩散模型的核心。我们的结果为这些作为CTRW驱动底层模型连续极限提供了严格的泛函刻画。针对严格M1收敛的移动平均和相关CTRW，我们发现随机积分的收敛性可能出现显著失效。然而，我们仍能识别出使M1收敛成立的自然被积过程类。进一步证明，这些结果具有足够的一般性，可为移动平均驱动的某类随机延迟微分方程建立M1拓扑下的泛函极限定理。