AI 中文总结
本文构建泊松嵌入标记Hawkes订单流模型，为含共同价格-波动率跳跃的粗糙Hawkes-Heston模型提供微观结构基础，推导其收敛性并经数值实验验证。
AI 中文摘要
El Euch等人（2018年，《Finance Stoch.》第22卷第2期，241-280页）开发了基于Hawkes的粗糙波动率、杠杆效应及粗糙Heston型极限的微观结构基础，并将其与El Euch和Rosenbaum（2019年，《Math. Finance》第29卷第1期，3-38页）的仿射粗糙Heston框架关联。Bondi等人（2024年，《Math. Finance》第34卷第4期，1197-1241页）提出的含共同价格-波动率跳跃的粗糙Hawkes-Heston模型，通过向粗糙仿射波动率添加状态依赖的共同跳跃扩展了该框架。我们通过构建泊松嵌入的标记Hawkes订单流模型，为其方差与共同跳跃机制提供微观结构基础：普通到达通过近不稳定重尾Hawkes机制产生粗糙连续波动率与杠杆效应，而罕见的标记到达代表共同冲击事件，会产生同步的价格跳跃与波动率激发。在近不稳定标度与简化形式容许条件下，完整重标后的价格/方差/跳跃系统沿完整序列收敛到唯一的完整正则粗糙Hawkes-Heston弱解。Hawkes更新结构产生Mittag-Leffler Volterra表示，随后被改写为Riemann-Liouville分数形式，极限系数以微观参数明确表示。该构建为粗糙Hawkes-Heston模型的方差与共同跳跃机制提供了微观结构基础，数值实验验证了我们的微观结构基础向粗糙Hawkes-Heston模型的收敛性。
英文摘要
Hawkes-based microstructural foundations for rough volatility, leverage, and rough Heston-type limits were developed by El Euch et al. (2018, Finance Stoch., 22(2), 241--280) and connected to the affine rough Heston framework of El Euch and Rosenbaum (2019, Math. Finance, 29(1), 3--38). The rough Hawkes--Heston model with common price--volatility jumps of Bondi et al. (2024, Math. Finance, 34(4), 1197--1241) extends this framework by adding state-dependent common jumps to rough affine volatility. We provide a microstructural foundation for its variance and common-jump mechanism by constructing a Poisson-embedded marked Hawkes order-flow model. Ordinary arrivals generate rough continuous volatility and leverage through a nearly unstable heavy-tailed Hawkes mechanism, while rare marked arrivals represent common shock events that produce simultaneous price jumps and volatility excitation. Under the nearly unstable scaling and the reduced-form admissibility conditions, the complete rescaled price/variance/jump system converges along the full sequence to the unique complete canonical rough Hawkes--Heston weak solution. The Hawkes renewal structure yields a Mittag--Leffler Volterra representation, which is then rewritten in Riemann--Liouville fractional form. The limiting coefficients are expressed explicitly in terms of the microscopic parameters. The construction provides a microstructural foundation for the variance and common-jump mechanism of the rough Hawkes--Heston model. Numerical experiments illustrate the convergence of our microstructural foundation to the rough Hawkes-Heston model.
Comments67 pages, 2 figures
