Paper: arXiv 2506.11813

Authors: Etienne Chevalier, Yadh Hafsi, Vathana Ly Vath, Sergio Pulido

Abstract

We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton-Jacobi-Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematical machinery, including singular stochastic control, Hamilton-Jacobi-Bellman variational inequalities, viscosity solutions, and regime-switching Markov chains, which is highly theoretical and dense. While it presents numerical examples, the methodology is primarily analytical, lacking the backtest-ready empirical validation, datasets, or statistical metrics typically found in high-rigor finance papers.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Determine optimal execution strategy under liquidity uncertainty"] --> B["Methodology<br>Stochastic Control & Singular Control"]
  B --> C["Data/Inputs<br>Limit-Order Book Shapes & Liquidity Regimes"]
  C --> D["Computational Process<br>Solve HJB Variational Inequalities & Analyze Free Boundaries"]
  D --> E["Key Findings/Outcomes<br>Unique Viscosity Solution & Optimal Execution Timing"]

Cited References (102)

  1. Hawkes Processes in Finance (2015) — link
  2. Dynamic optimal execution in a mixed-market-impact Hawkes price model (2016) — link
  3. Optimal trading strategy and supply/demand dynamics (2013) — link
  4. Point Processes and Queues. Martingale Dynamics. Springer-Verlag, Berlin – Heidelberg – New York 1981, 373 S., 31 Abb., DM 88,– (1988) — link
  5. Exponential Martingales and Changes of Measure for Counting Processes (2015) — link
  6. Optimal reinsurance via BSDEs in a partially observable model with jump clusters (2023) — link
  7. Fundamentals of Stochastic Filtering (2008)
  8. Mean field approach to stochastic control with partial information (2021) — link
and 94 more
  1. Optimal investment in markets with over and under-reaction to information (2017) — link
  2. Limit Theorems for Stochastic Processes (2013)
  3. Stochastic Integration and Differential Equations (2005)
  4. Uncovering Market Disorder and Liquidity Trends Detection — → 2310.09273
  5. Optimal liquidation under partial information with price impact (2020) — link
  6. Optimal execution with multiplicative price impact and incomplete information on the return (2023) — link
  7. Optimal Control for Partially Observed Diffusions (1982) — link
  8. Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation (2020) — link
  9. Liquidity risk and optimal dividend/investment strategies (2017) — link
  10. A mixed singular/switching control problem for a dividend policy with reversible technology investment (2008) — link
  11. Applied Stochastic Control of Jump Diffusions (2007) — link
  12. Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control (2022) — link
  13. Trades, Quotes and Prices: Financial Markets Under the Microscope (2018)
  14. Slow Decay of Impact in Equity Markets (2015) — link
  15. Optimal execution strategies in limit order books with general shape functions (2010) — link
  16. How Markets Slowly Digest Changes in Supply and Demand (2009) — link
  17. Optimal Execution in a General One-Sided Limit-Order Book (2011) — link
  18. Separation principle for impulse control with partial information (1983) — link
  19. Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control (2022) — link
  20. Filtering of a Markov Jump Process with Counting Observations (2000) — link
  21. Sequential tracking of a hidden Markov chain using point process observations (2009) — link
  22. Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels (2019)
  23. Markov-modulated Hawkes processes for modeling sporadic and bursty event occurrences in social interactions (2022) — link
  24. An Introduction to the Theory of Point Processes (2007)
  25. Scaling limits for Hawkes processes and application to financial statistics (2012) — link
  26. Optimal control of execution costs (1998) — link
  27. Optimal execution of portfolio transactions (2001) — link
  28. Optimal Execution Cost for Liquidation Through a Limit Order Market (2013) — link
  29. Portfolio liquidation games with self-exciting order flow (2022) — link
  30. Second order BSDE under monotonicity condition and liquidation problem under uncertainty * (2017) — link
  31. Optimal High-Frequency Trading in a Pro Rata Microstructure with Predictive Information (2015) — link
  32. Optimal Trade Execution in Illiquid Markets (2011) — link
  33. Optimal execution with stochastic delay (2023) — link
  34. Optimal Trading with Stochastic Liquidity and Volatility (2012) — link
  35. A model of optimal portfolio selection under liquidity risk and price impact (2007) — link
  36. Portfolio optimisation with strictly positive transaction costs and impulse control (1998) — link
  37. Impulse control of piecewise-deterministic processes (1989) — link
  38. Doubly Stochastic Poisson-Processes (1991) — link
  39. The Dirichlet problem for the convex envelope (2010) — link
  40. Stability of nonlinear Hawkes processes (1996) — link
  41. Stochastic equations, flows and measure-valued processes (2012) — link
  42. Topological Spaces: Including a Treatment of Multi-valued Functions, Vector Spaces and Convexity (1963)
  43. Multilayer feedforward networks are universal approximators (1989) — link
  44. OPTIMAL HIGH-FREQUENCY TRADING IN A PRO RATA MICROSTRUCTURE WITH PREDICTIVE INFORMATION (2015) — link
  45. Optimal Portfolio Liquidation with Limit Orders (2012) — link
  46. Optimal Execution with Identity Optionality (2022) — link
  47. Optimal Execution with Rough Path Signatures (2020) — link
  48. The Separation Principle for Impulse Control Problems (1981)
  49. Mixed control problem under partial observation (1993) — link
  50. Finite Horizon Decision Timing with Partially Observable Poisson Processes (2012) — link
  51. Trade-throughs: Empirical Facts and Application to Lead-lag Measures (2011) — link
  52. Funktionentheorie (1992)
  53. Universal price impact functions of individual trades in an order-driven market (2012) — link
  54. Linear models for the impact of order flow on prices. I. History dependent impact models (2018) — link
  55. High Frequency Trading in a Limit Order Book (2008) — link
  56. No-dynamic-arbitrage and market impact (2010) — link
  57. Optimal Control of Trading Algorithms: A General Impulse Control Approach (2011) — link
  58. Optimal Execution under Incomplete Information — → 2411.04616
  59. Portfolio Selection with Transaction Costs (1990)
  60. Weak Dynamic Programming Principle for Viscosity Solutions (2011) — link
  61. Optimal portfolio execution problem with stochastic price impact (2020) — link
  62. User’s Guide to Viscosity Solutions of Second Order Artial Differential Equations (1992) — link
  63. Optimal stopping problem for jump–diffusion processes with regime-switching (2021) — link
  64. From the Optimal Singular Stochastic Control to the Optimal Stopping for Regime-Switching Processes (2023) — link
  65. Optimal stopping of switching diffusions with state dependent switching rates (2016) — link
  66. Optimal liquidation under stochastic liquidity (2018) — link
  67. Numerical Methods of Stochastic Control Problems in Continuous Time (1992) — link
  68. A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models (2005) — link
  69. Optimal Execution with Multiplicative Price Impact (2015) — link
  70. Optimal trade execution in order books with stochastic liquidity (2019) — link
  71. Optimal Trading with Signals and Stochastic Price Impact (2022) — link
  72. Optimal Trade Execution in an Order Book Model with Stochastic Liquidity Parameters (2021) — link
  73. Optimal execution with regime-switching market resilience (2019) — link
  74. Multi-dimensional optimal trade execution under stochastic resilience (2019) — link
  75. Partial Differential Equations (2022)
  76. Variational analysis (2009)
  77. The regularity of free boundaries in higher dimensions (1977) — link
  78. On the fine structure of the free boundary for the classical obstacle problem (2019) — link
  79. Numerical Solution of Partial Differential Equations: An Introduction (2005)
  80. Optimal investment and consumption with transaction costs (1994)
  81. A stochastic representation theorem with applications to optimization and obstacle problems (2004)
  82. Optimal stopping, free boundary, and American option in a jump-diffusion model (1997)
  83. Inequalities (1952)
  84. Principles of Mathematical Analysis (1976)
  85. Convex Analysis (1970)
  86. Stochastic Liquidity as a Proxy for Nonlinear Price Impact (2024) — link
  87. Portfolio choice with small temporary and transient price impact (2019) — link
  88. Trading with Small Nonlinear Price Impact (2020)
  89. Trading with Small Price Impact (2017) — link
  90. Hedging with temporary price impact (2017)
  91. Power Laws in Economics: An Introduction (2016)
  92. Selling a large stock position: a stochastic control approach with state constraints (2007)
  93. Optimal Liquidation of a Basket of Stocks using Reinforcement Learning (2023) — link
  94. Trading with Concave Price Impact and Impact Decay—Theory and Evidence (2025) — link

2 of 102 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.