In this paper, we study so-called general compound and regime-switching general compound Hawkes processes to model the price processes in the limit order books. We prove Law of Large Numbers (LLNs) and Functional Central Limit Theorems (FCLTs), the main results of the present paper, for both cases, non-regime-switching (Lemma 1 and Theorem 1) and regime-switching (Lemma 2 and Theorem 2) cases. The latter two FCLTs are applied to limit order books, where we use these asymptotic methods to study the link between price volatility and order flow in our two models by investigating the diffusion limits of these price processes. The volatilities of price changes are expressed in terms of parameters describing the arrival rates and price changes. Numerical examples are presented for LOBster and Cisco data.

在本文中，我们研究了所谓的一般复合和政权转换的一般复合Hawkes过程，以对限价订单簿中的价格过程进行建模。我们证明了大数定律（LLNs）和泛函中心极限定理（FCLTs），这是本文的主要结果，对于两种情况，非体制转换（引理1和定理1）和体制转换（引理2和定理2）例。后两个FCLT应用于限价订单簿，在这里我们使用这些渐近方法通过研究这些价格过程的扩散极限来研究两个模型中价格波动与订单流之间的联系。价格变化的波动性用描述到达率和价格变化的参数表示。给出了LOBster和Cisco数据的数值示例。