  In this paper we introduce two new Hawkes processes, namely, compound and
regime-switching compound Hawkes processes, to model the price processes in
limit order books. We prove Law of Large Numbers and Functional Central Limit
Theorems (FCLT) for both processes. The 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 using 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. We also present
some numerical examples.
