标题： 波动聚集的子过程模型分析：VGSA及其扩展 显示英文标题

标题： Analysing Models for Volatility Clustering with Subordinated Processes: VGSA and Beyond

摘要： 本文探讨了一类通过将布朗运动与 Levy 过程进行次序变换构建的全面时间变换随机过程，其中次序变换进一步由诸如 Cox Ingersoll Ross (CIR) 和 Chan Karolyi Longstaff Sanders (CKLS) 过程等随机到达机制所控制。 这些模型扩展了经典的跳跃框架，如方差伽马 (VG) 和 CGMY 过程，允许对市场特征如跳跃聚集、厚尾和波动持久性进行更灵活的建模。 我们首先回顾 Levy 次序变换器的理论，并在伽马次序下建立 VG 过程的强一致性结果。 在此基础上，我们证明当到达过程遵循 CIR 或 CKLS 动力学时，VG 和 VGSA（带有随机到达的 VG）过程的渐近正态性。 随后，分析扩展到更一般的 CGMY 过程在随机到达下的情况，在到达过程满足正性和规则性条件的情况下，我们推导出类似的相容性和极限定理。 一项模拟研究伴随着理论工作，通过蒙特卡洛实验验证了我们的结果，可视化和正态性检验（通过 Shapiro-Wilk 统计量）显示，即使对于由厚尾跳跃驱动的过程，也表现出近似高斯行为。 本研究为分析具有随机时间变化的次序模型提供了一个严格且统一的概率框架，适用于不确定性下的金融建模和推断。 显示英文摘要

摘要： This paper explores a comprehensive class of time-changed stochastic processes constructed by subordinating Brownian motion with Levy processes, where the subordination is further governed by stochastic arrival mechanisms such as the Cox Ingersoll Ross (CIR) and Chan Karolyi Longstaff Sanders (CKLS) processes. These models extend classical jump frameworks like the Variance Gamma (VG) and CGMY processes, allowing for more flexible modeling of market features such as jump clustering, heavy tails, and volatility persistence. We first revisit the theory of Levy subordinators and establish strong consistency results for the VG process under Gamma subordination. Building on this, we prove asymptotic normality for both the VG and VGSA (VG with stochastic arrival) processes when the arrival process follows CIR or CKLS dynamics. The analysis is then extended to the more general CGMY process under stochastic arrival, for which we derive analogous consistency and limit theorems under positivity and regularity conditions on the arrival process. A simulation study accompanies the theoretical work, confirming our results through Monte Carlo experiments, with visualizations and normality testing (via Shapiro-Wilk statistics) that show approximate Gaussian behavior even for processes driven by heavy-tailed jumps. This work provides a rigorous and unified probabilistic framework for analyzing subordinated models with stochastic time changes, with applications to financial modeling and inference under uncertainty.