General Compound Hawkes Processes in Limit Order Books

In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process.

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Multivariate General Compound Point Processes in Limit Order Books

Risks ◽

10.3390/risks8030098 ◽

2020 ◽

Vol 8 (3) ◽

pp. 98

Author(s):

Qi Guo ◽

Bruno Remillard ◽

Anatoliy Swishchuk

Keyword(s):

Point Process ◽

Point Processes ◽

Hawkes Process ◽

Limit Order ◽

Large Numbers ◽

Order Books ◽

Limit Order Books ◽

Functional Central Limit Theorems ◽

Multivariate Point ◽

Functional Central Limit

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. The law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data.

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A Law of Large Numbers for Limit Order Books

Mathematics of Operations Research ◽

10.1287/moor.2017.0848 ◽

2017 ◽

Vol 42 (4) ◽

pp. 1280-1312 ◽

Cited By ~ 13

Author(s):

Ulrich Horst ◽

Michael Paulsen

Keyword(s):

Law Of Large Numbers ◽

Limit Order ◽

Large Numbers ◽

Order Books ◽

Limit Order Books

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Compound Hawkes Processes in Limit Order Books

SSRN Electronic Journal ◽

10.2139/ssrn.2987943 ◽

2017 ◽

Cited By ~ 2

Author(s):

Anatoliy V. Swishchuk ◽

Jonathan Chavez-Casillas ◽

Robert Elliott

Keyword(s):

Limit Order ◽

Hawkes Processes ◽

Order Books ◽

Limit Order Books

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Order Book Dynamics with Liquidity Fluctuations: Limit Theorems and Large Deviations

10.20944/preprints202106.0478.v1 ◽

2021 ◽

Author(s):

Helder Rojas ◽

Anatoly Yambartsev ◽

Artem Logachov

Keyword(s):

Large Deviations ◽

Stochastic Models ◽

Limit Theorems ◽

Law Of Large Numbers ◽

Limit Order Book ◽

Order Book ◽

Limit Order ◽

Bid Ask Spread ◽

Large Numbers ◽

The One

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

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