Paper: arXiv 2510.26438

Authors: Konark Jain, Nick Firoozye, Jonathan Kochems, Philip Treleaven

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 7.8/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematical frameworks including high-dimensional non-local HJB-QVIs and impulse control theory, indicating high mathematical complexity. It demonstrates strong empirical rigor by validating its models through a high-fidelity Hawkes LOB simulator, comparing RL and deep learning solvers, and reporting specific performance metrics like Sharpe ratios.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Optimal Market Making<br>in Hawkes LOB Market"] --> B["Data & Market Model"]
  B --> C["Problem Formulation<br>Impulse Control HJB-QVI"]
  C --> D{"Computational Approach"}
  D --> E["RL Approximation<br>PPO + Self-Imitation"]
  D --> F["Deep Learning Method<br>HJB Solver"]
  E & F --> G["Performance Comparison<br>Sharpe Ratio > 30"]
  G --> H["Key Findings<br>RL outperforms traditional methods<br>in high-dimensional impulse control"]

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  316. On Optimal Tracking Portfolio in Incomplete Markets : The Reinforcement Learning Approach — link
  317. Recent developments in machine learning methods for stochastic control and games (2023) — link
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  324. A theory of passive market impact — → 2412.07461
  325. The two square root laws of market impact and the role of sophisticated market participants — → 2311.18283
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  328. Limit theorems for nearly unstable Hawkes processes (2015) — link
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  349. Robust Market Making via Adversarial Reinforcement Learning (2020) — link
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  515. HYDRODYNAMIC LIMIT OF ORDER - BOOK DYNAMICS (2018) — link
  516. The predictive power of zero intelligence in financial markets
  517. ALGORITHMIC TRADING WITH MARKOV CHAINS
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  534. Stochastic systems with memory and jumps (2019) — link
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  551. The probability of backtest overfitting (2016)
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  553. The role of volume in order book dynamics: a multivariate Hawkes process analysis (2017)
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  555. Econophysics review: II . Agent -based models (2011)
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  557. Hawkes model specification for limit order books (2022)
  558. Limit Order Books (2016) — link
  559. Conditional Generators for Limit Order Book Environments : Explainability , Challenges , and Robustness — → 2306.12806
  560. Probability distribution of returns in the Heston model with stochastic volatility* (2002) — link
  561. Hawkes Process : Fast Calibration , Application to Trade Clustering , and Diffusive Limit (2014) — link
  562. Learning to simulate realistic limit order book markets from data as a World Agent (2022)
  563. Deep Learning (2016)
  564. Marked Hawkes process modeling of price dynamics and volatility estimation (2017)
  565. Generative AI for End -to- End Limit Order Book Modelling : A Token - Level Autoregressive Generative Model of Message Flow Using a Deep State Space Network — → 2309.00638
  566. Deep learning for market by order data (2021)
  567. Neural Stochastic Agent - Based Limit Order Book Simulation : A Hybrid Methodology — → 2303.00080
  568. The neural hawkes process: A neurally self-modulating multivariate point process (2017)
  569. Fast agent-based simulation framework of limit order books with applications to pro-rata markets and the study of latency effects (2020) — link
  570. State Dependent Parallel Neural Hawkes Process for Limit Order Book Event Stream Prediction and Simulation (2022)
  571. The Limit Order Book Recreation Model ( LOBRM ): An Extended Analysis (2021)
  572. The lob recreation model: Predicting the limit order book from taq history using an ordinary differential equation recurrent neural network (2021)
  573. Limit order book as a market for liquidity (2005)
  574. Price discovery and learning during the preopening period in the Paris Bourse (1999)
  575. The price impact of order book events (2014)
  576. Generative adversarial nets (2014)
  577. Intra- Day Price Simulation with Generative Adversarial Modelling of the Order Flow (2021)
  578. Dynamic calibration of order flow models with generative adversarial networks (2022)
  579. Order book dynamics with liquidity fluctuations: limit theorems and large deviations (2020) — link
  580. Wasserstein generative adversarial networks (2017)
  581. Conditional generative adversarial nets (2014) — link
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  583. Dynamic equilibrium limit order book model and optimal execution problem (2014) — link
  584. Hawkes-based models for high frequency financial data (2022)
  585. Modeling Bid and Ask Price Dynamics with an Extended Hawkes Process and Its Empirical Applications for High - Frequency Stock Market Data (2022)
  586. Order-book modeling and market making strategies (2018)
  587. Equilibrium Model of Limit Order Books : A Mean - Field Game View (2022)
  588. A stochastic partial differential equation model for limit order book dynamics (2021)
  589. Second- Order Approximation of Limit Order Books in a Single - Scale Regime — → 2308.00805
  590. Second order approximations for limit order books (2018)
  591. Algorithmic trading with Markov chains (2010)
  592. Limit order books, diffusion approximations and reflected SPDEs : from microscopic to macroscopic models (2020)
  593. A one-level limit order book model with memory and variable spread (2017)
  594. Estimation of an Order Book Dependent Hawkes Process for Large Datasets — → 2307.09077
  595. Ergodicity and diffusivity of Markovian order book models: a general framework (2017)
  596. A weak law of large numbers for a limit order book model with fully state dependent order dynamics (2017)
  597. Deep learning modeling of limit order book: A comparative perspective (2020) — link
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  600. A law of large numbers for limit order books (2017)
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  605. Hawkes processes and their applications to finance: a review (2018) — link
  606. From asymptotic properties of general point processes to the ranking of financial agents (2019) — link
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  608. Measuring the resiliency of an electronic limit order book (2007) — link
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  610. Estimation of slowly decreasing hawkes kernels: application to high-frequency order book dynamics (2016)
  611. Ergodicity and scaling limit of a constrained multivariate Hawkes process (2014) — link
  612. Hawkes Processes in Finance (2015) — link
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  614. Order book dynamics in liquid markets: limit theorems and diffusion approximations (2011) — link
  615. An agent based model of the E - Mini S & P 500 applied to flash crash analysis (2012) — link
  616. A Mathematical Approach to Order Book Modelling (2011) — link
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  618. Statistical Modeling of High - Frequency Financial Data (2011) — link
  619. Modelling high-frequency limit order book dynamics with support vector machines (2015) — link
  620. Simulating and Analyzing Order Book Data : The Queue - Reactive Model (2015) — link
  621. A steady-state model of the continuous double auction (2003) — link
  622. Universal features of price formation in financial markets: perspectives from Deep Learning (2018)
  623. Quant GANs : deep generation of financial time series (2020) — link
  624. DeepLOB : Deep Convolutional Neural Networks for Limit Order Books (2019) — link
  625. Limit order books (2013) — link
  626. The Price Impact of Order Book Events (2013) — link
  627. The predictive power of zero intelligence in financial markets (2005) — link
  628. Statistical theory of the continuous double auction (2003) — link
  629. A Stochastic Model for Order Book Dynamics (2010) — link
  630. An Ephemerally Self - Exciting Point Process (2021) — link
  631. An Ephemerally Self - Exciting Point Process (2021) — link
  632. State-dependent Hawkes processes and their application to limit order book modelling (2022) — link
  633. A single queue-reactive Hawkes model for the order flow (2023) — link
  634. Simulating and Analyzing Order Book Data : The Queue - Reactive Model (2015) — link
  635. Algorithmic trading in a microstructural limit order book model (2020) — link
  636. Diffusion Approximations for Self -excited Systems with Applications to General Branching Processes (2024) — link
  637. Order Book Queue Hawkes Markovian Modeling (2024) — link
  638. A Scaling Limit for Limit Order Books Driven by Hawkes Processes (2019) — link
  639. A Mathematical Framework for Modelling Order Book Dynamics — → 2302.01169
  640. Long- Time Behavior of a Hawkes Process – Based Limit Order Book (2015) — link
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  642. Exogenous and endogenous price jumps belong to different dynamical classes (2022) — link
  643. Modelling systemic price cojumps with Hawkes factor models (2015) — link
  644. The endo–exo problem in high frequency financial price fluctuations and rejecting criticality (2019) — link
  645. Hawkes process model with a time-dependent background rate and its application to high-frequency financial data (2017) — link
  646. Markets, Herding and Response to External Information (2015) — link
  647. Modelling illiquidity spillovers with Hawkes processes: an application to the sovereign bond market (2018) — link
  648. Detection of intensity bursts using Hawkes processes: An application to high-frequency financial data (2018) — link
  649. Robust identification of controlled Hawkes processes (2020) — link
  650. Modeling FX market activity around macroeconomic news: a Hawkes process approach (2015) — link
  651. Optimal Execution under Incomplete Information — → 2411.04616
  652. A tutorial on the deterministic Impulse Control Maximum Principle : Necessary and sufficient optimality conditions (2012) — link
  653. Limit Order Book Simulations: A Review (2024) — link
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  655. Malliavin calculus and optimal control of stochastic Volterra equations (2015)
  656. An introduction to computational stochastic PDEs (2014)

25 of 664 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.