We study the optimal Market Making problem in a Limit Order Book (LOB) market simulated using a high-fidelity, mutually exciting Hawkes process. Departing from traditional Brownian-driven mid-price models, our setup captures key microstructural properties such as queue dynamics, inter-arrival clustering, and endogenous price impact. Recognizing the realistic constraint that market makers cannot update strategies at every LOB event, we formulate the control problem within an impulse control framework, where interventions occur discretely via limit, cancel, or market orders. This leads to a high-dimensional, non-local Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJB-QVI), whose solution is analytically intractable and computationally expensive due to the curse of dimensionality. To address this, we propose a novel Reinforcement Learning (RL) approximation inspired by auxiliary control formulations. Using a two-network PPO-based architecture with self-imitation learning, we demonstrate strong empirical performance with limited training, achieving Sharpe ratios above 30 in a realistic simulated LOB. In addition to that, we solve the HJB-QVI using a deep learning method inspired by Sirignano and Spiliopoulos 2018 and compare the performance with the RL agent. Our findings highlight the promise of combining impulse control theory with modern deep RL to tackle optimal execution problems in jump-driven microstructural markets.

Tick-sizes not only influence the granularity of the price formation process but also affect market agents' behavior. We investigate the disparity in the microstructural properties of the Limit Order Book (LOB) across a basket of assets with different relative tick-sizes. A key contribution of this study is the identification of several stylized facts, which are used to differentiate between large, medium, and small-tick assets, along with clear metrics for their measurement. We provide cross-asset visualizations to illustrate how these attributes vary with relative tick-size. Further, we propose a Hawkes Process model that not only fits well for large-tick assets, but also accounts for sparsity, multi-tick level price moves, and the shape of the LOB in small-tick assets. Through simulation studies, we demonstrate the versatility of the model and identify key variables that determine whether a simulated LOB resembles a large-tick or small-tick asset. Our tests show that stylized facts like sparsity, shape, and relative returns distribution can be smoothly transitioned from a large-tick to a small-tick asset using our model. We test this model's assumptions, showcase its challenges and propose questions for further directions in this area of research.

Hawkes Process has been used to model Limit Order Book (LOB) dynamics in several ways in the literature however the focus has been limited to capturing the inter-event times while the order size is usually assumed to be constant. We propose a novel methodology of using Compound Hawkes Process for the LOB where each event has an order size sampled from a calibrated distribution. The process is formulated in a novel way such that the spread of the process always remains positive. Further, we condition the model parameters on time of day to support empirical observations. We make use of an enhanced non-parametric method to calibrate the Hawkes kernels and allow for inhibitory cross-excitation kernels. We showcase the results and quality of fits for an equity stock's LOB in the NASDAQ exchange and compare them against several baselines. Finally, we conduct a market impact study of the simulator and show the empirical observation of a concave market impact function is indeed replicated.

Limit Order Books (LOBs) serve as a mechanism for buyers and sellers to interact with each other in the financial markets. Modelling and simulating LOBs is quite often necessary for calibrating and fine-tuning the automated trading strategies developed in algorithmic trading research. The recent AI revolution and availability of faster and cheaper compute power has enabled the modelling and simulations to grow richer and even use modern AI techniques. In this review we examine the various kinds of LOB simulation models present in the current state of the art. We provide a classification of the models on the basis of their methodology and provide an aggregate view of the popular stylized facts used in the literature to test the models. We additionally provide a focused study of price impact's presence in the models since it is one of the more crucial phenomena to model in algorithmic trading. Finally, we conduct a comparative analysis of various qualities of fits of these models and how they perform when tested against empirical data.

We propose a nonparametric method for simulating the dynamics of a limit order book using Generative Adversarial Networks (GAN) to learn the conditional distribution of the future state of the order book given its current state from time series of the limit order book. Our method yields a scenario generator for limit order books which captures a range of stylized facts and salient properties of limit order book transitions. We show that the trained generator is also able to correctly reproduce some key properties observed in empirical studies on market impact. In particular, the model exhibits a decaying marginal impact of trade size, higher impact of aggressive orders, as well as a decreasing relation between impact and order book depth.

Classical models for order flow dynamics based on point processes, such as Poisson or Hawkes processes, have been studied intensively. Often, several days of limit border book (LOB) data is used to calibrate such models, thereby averaging over different dynamics - such as intraday effects or different trading volumes. This work uses generative adversarial networks (GANs) to learn the distribution of calibrations – obtained by many calibrations based on short time frames. The trained GAN can then be used to generate synthetic, realistic calibrations based on external conditions such as time of the day or volatility. Results show that GANs easily reproduce patterns of the order arrival intensities and can fit the distribution well without heavy parameter tuning. The synthetic calibrations can then be used to simulate order streams which contain new dynamics such as temporary drifts, different volatility regimes, but also intra-day patterns such as the commonly observed U-shape that reflects stylized behaviour around open and close of market hours.

This article discusses a new application of reinforcement learning: to the problem of hedging a portfolio of “over-the-counter” derivatives under under market frictions such as trading costs and liquidity constraints. It is an extended version of our recent work https://www.ssrn.com/abstract=3120710, here using notation more common in the machine learning literature. The objective is to maximize a non-linear risk-adjusted return function by trading in liquid hedging instruments such as equities or listed options. The approach presented here is the first efficient and model-independent algorithm which can be used for such problems at scale.

We outline the idiosyncrasies of neural information processing and machine learning in quantitative finance. We also present some of the approaches we take towards solving the fundamental challenges we face.

We show the diagonal problem for higher-order pushdown automata (HOPDA), and hence the simultaneous unboundedness problem, is decidable. From recent work by Zetzsche this means that we can construct the downward closure of the set of words accepted by a given HOPDA. This also means we can construct the downward closure of the Parikh image of a HOPDA. Both of these consequences play an important role in verifying concurrent higher-order programs expressed as HOPDA or safe higher-order recursion schemes.

Asynchronously communicating pushdown systems (ACPS) that satisfy the empty-stack constraint (a pushdown process may receive only when its stack is empty) are a popular decidable model for recursive programs with asynchronous atomic procedure calls. We study a relaxation of the empty-stack constraint for ACPS that permits concurrency and communication actions at any stack height, called the shaped stack constraint, thus enabling a larger class of concurrent programs to be modelled. We establish a close connection between ACPS with shaped stacks and a novel extension of Petri nets: Nets with Nested Coloured Tokens (NNCTs). Tokens in NNCTs are of two types: simple and complex. Complex tokens carry an arbitrary number of coloured tokens. The rules of NNCT can synchronise complex and simple tokens, inject coloured tokens into a complex token, and eject all tokens of a specified set of colours to predefined places. We show that the coverability problem for NNCTs is Tower-complete. To our knowledge, NNCT is the first extension of Petri nets, in the class of nets with an infinite set of token types, that has primitive recursive coverability. This result implies Tower-completeness of coverability for ACPS with shaped stacks.

This paper presents an approach to verify safety properties of Erlang-style, higher-order concurrent programs automatically. Inspired by Core Erlang, we introduce λ Actor, a prototypical functional language with pattern-matching algebraic data types, augmented with process creation and asynchronous message-passing primitives. We formalise an abstract model of λ Actor programs called Actor Communicating System (ACS) which has a natural interpretation as a vector addition system, for which some verification problems are decidable. We give a parametric abstract interpretation framework for λ Actor and use it to build a polytime computable, flow-based, abstract semantics of λ Actor programs, which we then use to bootstrap the ACS construction, thus deriving a more accurate abstract model of the input program.

In this paper, we study the program-point reachability problem of concurrent pushdown systems that communicate via unbounded and unordered message buffers. Our goal is to relax the common restriction that messages can only be retrieved by a pushdown process when its stack is empty. We use the notion of partially commutative context-free grammars to describe a new class of asynchronously communicating pushdown systems with a mild shape constraint on the stacks for which the program-point coverability problem remains decidable. Stacks that fit the shape constraint may reach arbitrary heights; further a process may execute any communication action (be it process creation, message send or retrieval) whether or not its stack is empty. This class extends previous computational models studied in the context of asynchronous programs, and enables the safety verification of a large class of message passing programs.

This paper presents Soter, a fully-automatic program analyser and verifier for Erlang modules. The fragment of Erlang accepted by Soter includes the higher-order functional constructs and all the key features of actor concurrency, namely, dynamic and possibly unbounded spawning of processes and asynchronous message passing. Soter uses a combination of static analysis and infinite-state model checking to verify safety properties specified by the user. Given an Erlang module and a set of properties, Soter first extracts an abstract (approximate but sound) model in the form of an actor communicating system (ACS), and then checks if the properties are satisfied using a Petri net coverability checker, BFC. To our knowledge, Soter is the first fully-automatic, infinite-state model checker for a large fragment of Erlang. We find that in practice our abstraction technique is accurate enough to verify an interesting range of safety properties such as mutual-exclusion and boundedness of mailboxes. Though the ACS coverability problem is Expspace-complete, Soter can analyse these problems surprisingly efficiently.

The collecting semantics of a program defines the strongest static property of interest. We study the analysis of the collecting semantics of higher-order functional programs, cast as left-linear term rewriting systems. The analysis generalises functional flow analysis and the reachability problem for term rewriting systems, which are both undecidable. We present an algorithm that uses indexed linear tree grammars (ILTGs) both to describe the input set and compute the set that approximates the collecting semantics. ILTGs are equi-expressive with pushdown tree automata, and so, strictly more expressive than regular tree grammars. Our result can be seen as a refinement of Jones and Andersen's procedure, which uses regular tree grammars. The main technical innovation of our algorithm is the use of indices to capture (sets of) substitutions, thus enabling a more precise binding analysis than afforded by regular grammars. We give a simple proof of termination and soundness, and demonstrate that our method is more accurate than other approaches to functional flow and reachability analyses in the literature.