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arxiv: 2506.11813 · v2 · submitted 2025-06-13 · 💱 q-fin.MF · q-fin.TR

Optimal Execution under Liquidity Uncertainty

Etienne Chevalier , Yadh Hafsi , Vathana Ly Vath , Sergio Pulido This is my paper

Pith reviewed 2026-05-19 10:05 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.TR
keywords optimal executionprice impactmarket resilienceviscosity solutionHamilton-Jacobi-Bellmansingular controlfree boundaryregime switching
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The pith

The value function for optimal share execution under stochastic resilience and liquidity regimes is the unique viscosity solution to a system of variational HJB inequalities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models the problem of buying a large block of shares while facing price impact that decays at a stochastic rate and liquidity that switches between regimes. It treats the trader's choices of timing and purchase rate as a singular stochastic control problem whose solution is characterized mathematically. The central result establishes that the associated value function satisfies a system of variational Hamilton-Jacobi-Bellman inequalities and is the unique viscosity solution of that system. This characterization also yields analytical properties of the free boundary that divides the region where it is optimal to execute from the region where it is optimal to wait. A reader cares because the result supplies a rigorous way to compute or approximate the least-cost schedule when market replenishment speed and liquidity levels are themselves random.

Core claim

For the optimal execution problem with general limit-order-book shapes, a stochastic volume-effect process governing the decay of price impact, and a finite-state Markov chain driving abrupt liquidity shifts, the value function is the unique viscosity solution to the system of variational Hamilton-Jacobi-Bellman inequalities. The free boundary separating the execution region from the continuation region admits further analytical study, and the model is illustrated by numerical examples for different book configurations.

What carries the argument

The system of variational Hamilton-Jacobi-Bellman inequalities satisfied by the value function, whose free boundary demarcates execution from continuation.

If this is right

  • The optimal purchase schedule can be approximated by solving the HJB system numerically for any fixed limit-order-book shape.
  • The location of the free boundary depends explicitly on the current resilience level and the active liquidity regime.
  • Abrupt shifts in the Markov chain produce jumps in the continuation value that alter the timing of optimal purchases.
  • General limit-order-book shapes enter the HJB system only through the instantaneous impact term, leaving the viscosity-solution argument unchanged.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same viscosity framework could be applied to the symmetric selling problem by reversing the sign of the control.
  • If the Markov chain is replaced by a continuous-state diffusion for liquidity, the system would become a single integro-differential HJB equation whose free boundary might still be characterizable.
  • The numerical examples suggest that execution becomes more front-loaded when the probability of moving to a low-liquidity regime rises.

Load-bearing premise

Market resilience decays according to the chosen stochastic volume-effect process and liquidity changes occur only by switching among the finite number of regimes in the Markov chain.

What would settle it

Numerical solution of the HJB system for a given resilience process and Markov chain produces an execution schedule whose realized cost, when simulated in a market whose liquidity dynamics differ markedly from the modeled processes, exceeds the cost of simple time-weighted averaging.

Figures

Figures reproduced from arXiv: 2506.11813 by Etienne Chevalier, Sergio Pulido, Vathana Ly Vath, Yadh Hafsi.

Figure 1
Figure 1. Figure 1: Representation of the limit order book at times [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Y t,y,X and Y t,y′ ,Xe sample paths in the scenario where y < y′ , τ < θ < T and ∆Xu + Y t,y,X u − Y t,y′ ,Xb u < X − Xb τ− . Therefore, we obtain the following inequality vi(t, x, y′ ) − vi(t, x, y) ≤ ε + R1 + R2 + R3 + R4 + R5 + R6, (10) where R1 = E h Z T t ψIu− (Yˇ t,y′ ,Xe u− )dXec u − Z T t ψIu− (Yˇ t,y,X u− )dXc u i ≤ 0, R2 = E h X t≤u<τ;∆Xu>0 ΦIu (Y t,y′ ,Xe u ) − ΦIu− (Yˇ t,y′ ,Xe … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of Oe and Oc. In this setting, define the function f0 : Oc ∪ Oe ∪ {(t0, x0, y1)} → R satisfying f0(t0, x0, y) := ( −∂ + x vi(t0, x0, y1) − ∂ − y vi(t0, x0, y) − ψi(y), if y < y1. −∂ + x vi(t0, x0, y1) − ∂ + y vi(t0, x0, y) − ψi(y), if y ≥ y1. (21) Given that f0(t0, x0, y1) = lim infy→y1 + f0(t0, x0, y) = 0, two scenarios may arise: lim inf y→y1 − f0(t0, x0, y) < η0 < 0, or lim inf y→y1 − f0(t0… view at source ↗
Figure 4
Figure 4. Figure 4: Variation of the value function over time under different market conditions with jump [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of continuation (blue) and exercise (red) regions, plotted against purchased [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of the exercise boundary x 7→ y ∗ (T /2, x) under different market conditions with jump intensity on the left, resilience in the middle, and volatility on the right. The exercise region expands in response to jumps. Indeed, when jump intensity is high, the optimal strategy involves anticipating increased future price impact by executing larger trades earlier. Conversely, when the volatility of th… view at source ↗
Figure 7
Figure 7. Figure 7: Variation of the exercise boundary x 7→ y ∗ (T /2, x) of v1 on the left and v2 on the right under different price impacts [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variation of the exercise boundary x 7→ y ∗ (T /2, x) of v1 on the left and v2 on the right under different regime switching intensities. A higher switching intensity expands the exercise region in the low price impact regime (see the left-hand side of [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Variation of the exercise boundary x 7→ y ∗ (T /2, x) of v1 on the left and v2 on the right under different asymmetric regime switching intensities [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
read the original 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. We allow for general limit order book shapes to characterize instantaneous market impact. To model the resilience dynamics, we introduce a stochastic process that governs the rate at which the deviation between the impacted and unaffected prices decays. This volume-effect process reflects fluctuations in market activity that drive the pace of liquidity replenishment. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies optimal execution of a large block purchase over a fixed horizon under general price impact that dissipates according to a stochastic resilience process (volume-effect process) and abrupt liquidity shifts modeled by a finite-state Markov chain. The trader solves a singular stochastic control problem to minimize costs by choosing timing and rate of purchases. The central claims are that the value function satisfies a system of variational Hamilton-Jacobi-Bellman inequalities, is the unique viscosity solution to this system, and that the free boundary separating execution and continuation regions admits analytical study; numerical examples under varying limit-order-book shapes illustrate the results.

Significance. If the viscosity uniqueness and free-boundary properties are established rigorously, the work extends singular-control techniques to execution problems with stochastic resilience and regime switches, providing a framework that captures realistic fluctuations in market activity and liquidity. This is relevant for algorithmic trading and market-microstructure modeling. The choice of viscosity solutions is natural for the singular-control setting with jumps in regimes, and the numerical illustrations help show the interplay among impact, resilience, and liquidity regimes.

major comments (1)
  1. [Abstract and §3] Abstract and §3 (Main results): the claim that the value function is the unique viscosity solution to the variational HJB system is load-bearing for the entire analysis, yet the provided outline does not include the verification step that the stochastic resilience process and Markov-chain liquidity dynamics satisfy the required regularity and growth conditions for the comparison principle; without this explicit check the uniqueness result cannot be confirmed from the given information.
minor comments (2)
  1. [Abstract] The abstract mentions numerical examples under different LOB configurations but does not specify the exact shapes, parameter values, or discretization scheme used; adding these details would improve reproducibility.
  2. [Model section] Notation for the resilience process and the regime-switching generator should be introduced with a dedicated table or list of symbols to avoid ambiguity when reading the HJB system.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (Main results): the claim that the value function is the unique viscosity solution to the variational HJB system is load-bearing for the entire analysis, yet the provided outline does not include the verification step that the stochastic resilience process and Markov-chain liquidity dynamics satisfy the required regularity and growth conditions for the comparison principle; without this explicit check the uniqueness result cannot be confirmed from the given information.

    Authors: We acknowledge the referee's observation. The uniqueness of the viscosity solution is indeed central to our results. While the manuscript invokes standard comparison principles for such HJB systems with regime switches, we agree that an explicit verification of the regularity and growth conditions for the stochastic resilience process and the Markov-chain liquidity dynamics should be provided to make the application of the comparison principle fully rigorous. In the revised manuscript, we will include this verification in Section 3, detailing how our assumptions ensure the necessary conditions hold. This addition will not alter the main claims but will enhance the completeness of the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard viscosity techniques

full rationale

The paper introduces a stochastic resilience process and finite-state Markov chain for liquidity, formulates the singular control problem, and applies standard techniques from stochastic control to show that the value function satisfies and is the unique viscosity solution to the system of variational HJB inequalities. The free-boundary analysis follows from the model dynamics without any reduction of claimed results to fitted parameters, self-citations, or definitional tautologies. All steps are independent of the target claims and rest on established theory for regime-switching singular control problems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard stochastic-control assumptions plus two domain-specific modeling choices for resilience and liquidity; no free parameters or invented entities are explicitly listed in the abstract.

axioms (2)
  • domain assumption Price impact dissipates according to a stochastic process governing the decay rate between impacted and unaffected prices.
    Introduced to reflect fluctuations in market activity and liquidity replenishment.
  • domain assumption Liquidity variations are captured by a regime-switching Markov chain.
    Used to model abrupt shifts in market conditions.

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