Abstract
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.
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