We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterised by the affine form of their cumulant-generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant-generating function of an AFI model satisfies a generalised convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to an AFV model.

中文翻译：

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仿射前向方差模型

我们介绍了仿射前向方差（AFV）模型类别，其中常规Heston模型和Rough Heston模型都是特例。我们表明，AFV模型可以通过其累积量生成函数的仿射形式来表征，该仿射形式可以作为卷积Riccati方程的解获得。我们进一步介绍仿射前向流动强度（AFI）模型，其结构与AFV模型相似，但受跳跃过程驱动，其中包括Hawkes型模型。我们表明，AFI模型的累积量生成函数满足广义卷积Riccati方程，并且AFI模型的高频极限在分布上收敛到AFV模型。