# General Compound Hawkes Processes in Limit Order Books

## TL;DR

This paper introduces new Hawkes process models for limit order book prices, deriving LLN and FCLT results to connect price volatility with order flow dynamics.

## Contribution

It develops and analyzes general compound and regime-switching Hawkes processes for modeling limit order book prices, providing new theoretical results.

## Key findings

- Derived LLN and FCLT for the proposed processes.
- Expressed price volatility in terms of order flow parameters.
- Linked price diffusion limits to order flow characteristics.

## Abstract

In this paper, we study various new Hawkes processes, namely, 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 (LLN) and Functional Central Limit Theorems (FCLT) for these processes. 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 studying 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.