#### About

I did a PhD in the Mathematics Department of Imperial College London under the supervision of Mikko Pakkanen. My research interests revolved around random processes and their applications to finance. More specifically, my research focused on limit order book modelling with interacting point processes. This is the website that I built to explain my research. You will find friendly introductions to my research topics as well as high-level summaries of my research findings. To know more about me, you can visit my LinkedIn profile.

##### SUMMARY

##### High-frequency financial data modelling with state-dependent Hawkes processes

Hawkes processes are a class of self-exciting point processes in which events of different types can precipitate each other, breaking the memorylessness property of Poisson processes. Given their ability to capture clustering phenomena, they have found numerous applications in finance in the last decade, in particular in the area of high-frequency data modelling. Indeed, the last twenty years have also seen the emergence of electronic order-driven markets, where agents submit buy and sell orders to a virtual exchange via their computers. As billions of orders are submitted each day, this brought a profusion of new data to study, with a chance to understand the price formation mechanism at the smallest timescales.

In this context, Hawkes processes have been used as a model of the order flow, the core idea being to specify a list of order types (i.e., orders with different effects on the market) and fit a Hawkes process to their timestamps to gain insights on the market dynamics. However, the main limitation of this approach is that Hawkes processes do not model the state of the market and its influence on the arrival rates of orders. That is why I have introduced the class of state-dependent Hawkes processes, an extension of Hawkes processes where the events can now interact with an auxiliary state process. I have worked on both the theoretical foundations of this new class (existence and uniqueness) and its application (statistical inference from real market data).

##### RESEARCH TOPICS

###### LIMIT ORDER BOOKS

Most of trading now happens on electronic markets through a limit order book mechanism. In short, the limit order book is the collection of awaiting buy and sell orders for a given stock. It can be understood as a snapshot of the instantaneous supply and demand. Market participants can observe it in real time and react to the incoming stream of orders. Finding adequate models is challenging and crucial for many applications.

###### MARKED POINT PROCESSES

A marked point process is a sequence of random times and random marks. In applications, the random times represent the arrival times of events while the random marks describe the events. For example, a mark equal to one (resp. two) will correspond to a buy (resp. sell) order. The intensity of a marked point process is a fundamental concept that quantifies how fast events are happening. It can depend on the history of the process.

###### HAWKES PROCESSES

Hawkes processes are a class of marked point processes that are defined in terms of their intensity. With these processes, an event of type A (e.g. a buy order) can precipitate an event of type B (e.g. a sell order). They are a typical example of interacting point process. For example, they are used in earthquake modelling, criminology, social network analysis and neurology. Not surprisingly, they also find many applications in finance.

##### RESEARCH FINDINGS

###### A NEW MODEL: HYBRID MARKED POINT PROCESSES

Hawkes or Markov?

On the one hand, limit order book models based on Hawkes processes account only for interactions between events (e.g. buy and sell orders) and ignore the current state of the limit order book (e.g. the spread). On the other hand, Markov models do capture the state of the limit order book, but the dynamics can only depend on the current state, meaning that the past has no influence and that interactions like in Hawkes processes are not possible.

A state process interacting with past-dependent events

Hybrid marked point processes generalise both Hawkes and Markov processes, addressing the dilemma mentioned above. A hybrid marked point process models both the state of a system and the arrival in time of events. Events can interact like in a Hawkes process, but their dynamics can now also depend on the state process. In parallel, as each event occurs, the state of the system transitions to a new value according to transition probabilities that vary with the event type. In short, a hybrid marked point process can account for complex interactions between a system and different event types.

Potential applications in the context of limit order books

High-frequency data analysis, statistical arbitrage, market impact assessment, algorithmic trading, optimal execution, market simulation.

Please contact me if you are interested in a collaboration.

###### EXISTENCE & UNIQUENESS VIA POISSON EMBEDDING

A hybrid marked point process is defined implicitly in terms of its intensity, which, in turn, depends on the history of the hybrid marked point process. Given the self-referential nature of this definition, it is not clear a priori that such a marked point process exists. One approach is to reformulate the existence problem as a well-known Poisson-driven stochastic differential equation. Existence of a solution to this equation will imply the existence of hybrid marked point processes. However, the existing techniques to solve this equation cannot be reused as they rely on assumptions that the intensity of a hybrid marked point process may fail to satisfy. This motivates the development of an alternative construction that can be applied to more general intensities. The solution is constructed piece by piece along the time axis, taking advantage of the discrete nature of the driving Poisson random measure.

###### ESTIMATION AND APPLICATION

I created the Python package mpoints, a machine learning tool that implements the class of state-dependent Hawkes processes. Its key features include both estimation (statistical inference) and simulation. Its application to market data has revealed that the endogeneity of the order flow does indeed depend on state variables like the spread and the queue imbalance.

##### PUBLICATIONS

###### HYBRID MARKED POINT PROCESSES: CHARACTERISATION, EXISTENCE AND UNIQUENESS

In collaboration with Mikko S. Pakkanen

Preprint, 21 July 2017

###### ON THE WEAK-HASH METRIC FOR BOUNDEDLY FINITE INTEGER-VALUED MEASURES

Bulletin of the Australian Mathematical Society. 2018:1-12. doi:10.1017/S0004972718000485.

###### STATE-DEPENDENT HAWKES PROCESSES AND THEIR APPLICATION TO LIMIT ORDER BOOK MODELLING

In collaboration with Mikko S. Pakkanen

Preprint, 21 September 2018

##### TALKS & POSTERS

###### London Graduate School PhD Day

Hybrid Marked Point Processes:

Characterisation, Existence and Uniqueness

London, 4 May 2018

###### Imperial-ETH Workshop on Mathematical Finance

Limit Order Book Modelling With State-Dependent Hawkes Processes

Zurich, 5 April 2018

###### German Probability and Statistics Days

Hybrid Marked Point Processes:

Characterisation, Existence and Uniqueness

Freiburg, 1 March 2018

###### Market Microstructure, The CFM-Imperial Workshop

Limit Order Book Modelling

With State-Dependent Hawkes Processes (poster)

London, 12 December 2017

###### Talk given to Hellebore Capital

Limit Order Book Modelling With State-Dependent Hawkes Processes

London, 28 November 2017

###### SIAM-JAMS Seminar, Imperial College London

Limit Order Book Modelling With Hybrid Marked Point Processes

London, 18 October 2017

###### CDT in Financial Computing and Analytics, PhD Talks, UCL

Limit Order Book Modelling With Hybrid Marked Point Processes

London, 4 October 2017

###### Conference on Ambit Fields and Related Topics, Aarhus University

Hybrid Marked Point Processes:

Characterisation, Existence and Uniqueness

Aarhus, 16 August 2017

###### Talk given to Investment Technology Group, Inc. (NYSE: ITG)

Limit Order Book Modelling: Core Concepts

London, 20 October 2016

###### SIAM-JAMS Seminar, Imperial College London

Limit Order Book Modelling With Interacting Point Processes

London, 4 October 2016

###### London-Paris Bachelier Workshop on Mathematical Finance

Reduced-Form Limit Order Book Modelling

With Point Processes (poster)

Paris, 30 September 2016

###### Conference on Ambit Fields and Related Topics, Aarhus University

Reduced-Form Limit Order Book Modelling

With Point Processes (poster)

Aarhus, 16 August 2016

###### At the Frontiers of Quantitative Finance, ICMS

Reduced-Form Limit Order Book Modelling With Point Processes

Edinburgh, 28 June 2016

###### CFM-Imperial Workshop on Quantitative Finance, CFM

Reduced-Form Limit Order Book Modelling With Point Processes

Paris, 13 June 2016

###### PhD Day, Imperial College London

From Level-1 Imbalance Measures To Deep Imbalance Measures

London, 16 December 2015

###### SIAM-JAMS Seminar, Imperial College London

Trading With Limit Order Books

London, 12 November 2015