FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts

Yang Li; Zhi Chen

REVIEW 4 major objections 5 minor 27 references

FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read FlowOE is a flow-matching imitation policy that learns from an ensemble of execution experts and claims to beat them through a refining adjustment.

desk verdict The claimed refining loss that lets FlowOE beat its experts is never defined; Algorithm 1 is the standard shortcut objective, so the central claim is unsupported despite a decent empirical study. read the letter →

arxiv 2506.05755 v1 pith:OFF4QUTK submitted 2025-06-06 cs.LG cs.AIcs.CEq-fin.CPq-fin.TR

Yang Li , Zhi Chen This is my paper

classification cs.LGcs.AIcs.CEq-fin.CPq-fin.TR

keywords optimalexecutionflowmatchingimitationlearningHestonstochasticvolatilityconcavemarketimpactshortcutpolicyAlmgren-Chrissmodelproximaloptimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

FlowOE is an imitation-learning framework for optimal execution: rather than solving the stochastic control problem from scratch, it trains a flow-matching generative policy to produce trading action sequences from demonstrations generated by established expert strategies—TWAP, VWAP, an Almgren-Chriss approximation, a Heston-optimal rule, and PPO agents. The market setting is a simulated Heston stochastic-volatility environment with concave temporary market impact $f(\nu)=\epsilon|\nu|^\beta$, which is more realistic than the constant-volatility, linear-impact assumptions of the classical model. The paper claims that this single policy adaptively selects the most suitable expert behavior for the prevailing market state and, through a refining loss and a fine-tuning adjustment, improves on the experts' own actions. Empirically, FlowOE is reported to achieve lower implementation shortfall, lower shortfall variance, and a lower Almgren-Chriss objective than every benchmark across high/low volatility and high/low impact scenarios, with the largest gains in the highly concave market ($\beta=0.3$). If correct, one flow-based policy could replace a collection of scenario-calibrated execution models and run at trading latency using a shortcut policy that acts in one to five steps.

What carries the argument

The central object is the Shortcut policy, a flow-matching generative model $s_\phi(a,t,\Delta t|\mathcal{O}_t)$ that outputs a velocity for the trading action $a$ conditioned on continuous time $t$, step size $\Delta t$, and market observation $\mathcal{O}_t=(t,X_t,S_t,V_t)$. It is trained so that a $2\Delta t$ step agrees with two $\Delta t$ steps (self-consistency, with stop-gradient targets) and so a fraction of each batch follows the straight rectified-flow target $x_1-x_0$, which is what lets the policy generate execution plans in 1–5 steps during inference. The environment it operates in is the Heston model, with price and variance driven by correlated Brownian motions and temporary impact $f(\nu)=\epsilon|\nu|^\beta$, so the expert actions being imitated already balance nonlinear impact cost against stochastic-volatility timing risk.

What would settle it

Inspect the released code and Algorithm 1: if no refining loss exists, train the same shortcut model with and without a separately added refinement step on the same expert dataset and compare the AC objective; if the with-refinement model shows no gain, the central outperformance claim is not supported by the stated mechanism.

Watch

Extended reading notes

Core claim

The paper claims that FlowOE, one flow-matching policy, learns the distribution of trading actions conditional on market state from demonstrations produced by several expert strategies, and that it then outperforms every expert it imitates. The policy is a 'shortcut' model trained so that one to five generation steps reproduce action sequences that normally require hundreds of ODE steps, with a direct rectified-flow target used for part of each batch and a self-consistency objective for the rest. In Monte Carlo trials across high/low volatility and high/low impact scenarios at three concavity levels, the authors report the lowest implementation shortfall, lowest standard deviation, and lowest Almgren-Chriss objective among TWAP, VWAP, AC-Approx, Heston-Optimal, and the shortcut model trained on PPO experts, including an up-to-10.9% shortfall reduction relative to AC-Approx and a 68% risk reduction. They attribute the gains to adaptive selection among expert behaviors and to a fine-tuning process that searches an adjustment action, which is added to the action generated by the imitation model.

Load-bearing premise

The load-bearing premise is that the refining loss and the fine-tuning adjustment described in the abstract and introduction are actually implemented in training and are what produce the outperformance over the experts, since the only training algorithm printed in the paper contains no such step.

Editorial extensions

If this is right

A single FlowOE policy can replace the zoo of scenario-calibrated expert models: the policy is trained once on diverse demonstrations and selects the appropriate behavior from the observed market state.
Flow-based execution becomes latency-feasible, because the shortcut policy generates the trading action sequence in one to five steps instead of hundreds of ODE solves.
In the most concave simulated market ($\beta=0.3$), the reported implementation shortfall improves by up to 10.9% over AC-Approx while the standard deviation of shortfall drops by about 68%.
Because the policy is an imitation of known experts, its behavior stays anchored to strategies that are already accepted in practice, which should make deployment and regulatory explanation easier than a black-box RL policy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the reported results are Monte Carlo trials inside the same Heston simulator used to generate the expert demonstrations, so the most valuable next test is to run FlowOE on historical order-flow data or in paper trading; the paper does not report such a test.
Editorial inference: the abstract and introduction credit a refining loss and a fine-tuning adjustment for the gains, but Algorithm 1 only shows the shortcut/consistency objective; retraining with and without a separately implemented refinement term would show whether the claimed mechanism is the real source of the improvement.
Editorial inference: the same flow-matching-plus-shortcut recipe transfers naturally to multi-asset execution and to other stochastic control problems with cheap-to-simulate experts, such as market making or inventory management.
Editorial inference: because the flow policy models a full conditional distribution rather than a point action, it could be used to generate multiple candidate execution plans and to estimate action uncertainty, which a deterministic expert trajectory cannot provide.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the headline claim is not supported by the paper's own method. FlowOE is supposed to improve on expert actions via a 'refining loss' and a fine-tuning step that searches an adjustment action. Neither appears in the body. Algorithm 1 is the standard shortcut-model objective from Frans et al., an l2 regression to the data direction plus a self-consistency target. There is no adjustment term, no second training phase, no objective that rewards beating the expert. So the mechanism credited for the results is absent.

What is actually there: a straightforward transfer of shortcut flow policies to optimal execution. That is new, and the simulation study is reasonably broad — Heston volatility, concave temporary impact, four scenarios, 10,000 trials per cell. The tables are arithmetically consistent (AC = IS + λ·STD² holds exactly), and the Shortcut Model trained on PPO demonstrations does achieve much lower AC in most cells. That is a real empirical pattern, and it suggests flow matching can act as a strong regularizer of expert policies. But that interpretation is not what the paper claims.

The soft spots are serious. First, the missing refining loss is load-bearing — without it, the 'improves upon the expert' claim is just unsupported. Second, the 'adaptively selects among diverse experts' claim is not tested: separate shortcut models are trained per expert and reported separately. Third, the Heston-Optimal expert (Eq. 17) is asserted with no derivation, and it is used both as a benchmark and as a training expert, so beating it is partly circular. Fourth, the parameter definitions for β are inconsistent: Table 1 lists {0.2, 0.6, 0.9}, the justification paragraph says {0.5, 1.0, 1.5}, and the tables use {0.5, 0.8, 0.3}. That is the kind of sloppiness that makes the experimental setup hard to trust. Also, Table 2 shows Shortcut Model HO with AC 412,934 against Heston-Optimal's 406,606 in HH, contradicting the 'consistently outperforms' sentence. No code, no confidence intervals, synthetic data only.

Bottom line: the empirical observation is not empty — the PPO-trained shortcut policy does look better on its training objective — but the paper's stated contribution, the refining mechanism, does not exist in the described method. Reframe it as 'flow matching regularizes expert demonstrations' and the results may hold. As written, it needs major revision. I'd send it to review, because the topic is relevant and the experiments are extensive enough to be worth refereeing, but I would not cite it in its current form.

Referee Report

4 major / 5 minor

Summary. The paper proposes FlowOE, an imitation-learning framework based on flow matching / shortcut models for optimal execution of a large sell order under Heston stochastic volatility and concave temporary market impact. Expert demonstrations are generated by traditional benchmarks (TWAP, VWAP, an approximate Almgren-Chriss strategy) plus a proposed 'Heston-Optimal' rule and PPO agents trained on a Heston simulator. A shortcut policy is trained on these expert trajectories, and the paper claims that a 'refining loss function' or fine-tuning process lets FlowOE not only imitate but also outperform the experts. Experiments report implementation shortfall (IS), standard deviation (STD), and an Almgren-Chriss objective (AC) across four volatility/impact scenarios and three β values, claiming consistent superiority of FlowOE over all benchmarks.

Significance. If the claims were substantiated, the paper would offer a practical application of generative flow-matching policies to optimal execution, with a potentially appealing latency reduction from many ODE steps to a few shortcut steps. However, the central contribution—the refining mechanism that allegedly yields performance beyond the learned experts—is never specified, and the empirical tables do not support the 'consistent outperformance' claim. The reported arithmetic in the tables (AC = IS + λ·STD²) is internally consistent, and the shortcut policy trained on PPO demonstrations does inherit PPO's strong AC values, which suggests the observed gains may be due to the quality of the PPO expert rather than any refinement step. As it stands, the paper provides neither a clear mechanism nor consistent evidence for its headline claim, so its significance is largely aspirational.

major comments (4)

[Abstract, Section 1, Algorithm 1] The abstract and Section 1 attribute FlowOE's outperformance to a 'refining loss function' and a 'fine-tuning process that searches an adjustment action', but Algorithm 1, the only training procedure in the paper, is the standard shortcut-model objective (ℓ2 regression toward x1−x0 for part of the batch and a stop-gradient self-consistency target for the rest). There is no adjustment-action term, no fine-tuning stage, and no objective that rewards beating the expert. The mechanism credited for the headline result is therefore absent from the manuscript, and the empirical gains are equally explained by the shortcut policy inheriting the performance of the PPO expert.
[Tables 2–4] The claim that FlowOE 'consistently and significantly outperformed the expert models' is contradicted by the reported numbers. In Table 2, Shortcut Model HO has a higher (worse) AC than Heston-Optimal in the HH scenario (412,934.98 vs 406,606.62) and the HL scenario (372,009.22 vs 366,023.67). The same pattern appears in Table 3 (HH: 415,259.48 vs 382,046.83; HL: 372,476.51 vs 325,037.99). Thus the 'Shortcut Model HO' often underperforms the very expert it was trained on, while the larger gains appear only for 'Shortcut Model PPO', which inherits PPO's superior AC values. The central claim of a single model that refines expert actions is not supported by these data.
[Eq. (17), Appendix B.4] The Heston-Optimal expert strategy in Eq. (17) is introduced without derivation from the stated optimal control problem, and it contains ad-hoc elements: the volatility-adjustment exponent 1/2, the factor (1+β), and the urgency cap in Appendix B.4 (min(2, 1/τ)). This invented expert is then used both as a training expert and as a benchmark, so the comparison is partly against a rule whose optimality is never established. Moreover, the paper does not implement a single adaptive model that selects among diverse experts; 'Shortcut Model HO' and 'Shortcut Model PPO' are separate shortcut policies trained on each expert, so the claim of 'adaptively selects the most suitable expert behavior' (abstract, Section 1) is not realized.
[Section 5.2, Tables 1, 2–4] The experimental setup is internally inconsistent regarding the market-impact nonlinearity parameter β. Section 5.2 states β∈{0.5,1.0,1.5}, Table 1 lists β∈{0.2,0.6,0.9}, and Tables 2–4 report results for β=0.5, 0.8, and 0.3. These conflicting specifications make the empirical results unreproducible and prevent the reader from knowing which β values actually generated the reported data.

minor comments (5)

[Section 6.2] The metric 'Implied Shortfall' should be 'Implementation Shortfall'; the paper uses 'Implied Shortfall' in the text and 'IS' in the tables, and the definition in Eq. (20) is for implementation shortfall, not implied shortfall.
[Throughout] There are numerous typos and grammatical issues (e.g., 'This paper propose flowOE', 'the final step (k=N−1) is liquidated: x_{N−1}=q(t_{N−1})' appears twice, inconsistent capitalization of 'flowOE' vs 'FlowOE'). A careful proofread is needed.
[Section 6.1, Eq. (17)] The Heston-Optimal trading rate in Eq. (17) is not dimensionally consistent: the term ((θ+(V(t)−θ)e^{−κ(T−t)})/V(t))^{1/2} is dimensionless, but the overall expression is multiplied by (1+β)q(t)/(T−t), which has units of shares per time; no derivation is provided for the exponent 1/2 or the (1+β) factor.
[Section 6.3, Table captions] The caption of Table 3 says 'β=0.8 (square-root impact model, less concave market)' and Table 4 says 'β=0.3 (square-root impact model, highly concave market)'; calling β=0.8 a 'square-root impact model' is misleading because square-root impact corresponds to β=0.5, and the text elsewhere uses different labels.
[Section 6, Figures] The figures (Figures 1–3) are described in text but appear to be missing from the manuscript; if they are included in the arXiv version, the captions should be self-contained and the standard-error shading explained.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity; the only self-citation is motivational, and the missing refinement mechanism is a support gap rather than a derivation that reduces to its inputs.

other [Section 1, paragraph beginning 'Flow matching models, recent generative modeling advancements...']
"This paradigm compellingly transfers to finance, particularly for optimal execution in stochastic environments. ... [16] first explore the application of an imitation learning framework to high-frequency trading tasks and demonstrates that flow matching-based models are capable of learning effectively within stochastic financial environments."

Reference [16] is the authors' own FlowHFT paper (Li, Chen & Yang), and it is the sole cited evidence that flow-matching imitation learning is effective in a stochastic financial setting. The present paper's premise that the robotics imitation-learning paradigm transfers to optimal execution therefore leans on a self-citation rather than on independent external support. This is motivational rather than derivation-forcing: Algorithm 1 and the experiments are self-contained, so the self-citation is minor and does not by itself force the paper's empirical conclusions.

full rationale

The claimed derivation chain is not circular in the mathematical sense. Algorithm 1 implements the standard shortcut/consistency objective, but the abstract and Section 1 promise a 'refining loss' and a 'fine-tuning process that searches an adjustment action' that do not appear in that algorithm; this is a serious missing-support and verifiability issue for the central outperformance claim, yet it is an unsupported assertion rather than an equation that reduces to its input. Similarly, the Heston-Optimal expert in Eq. 17 is asserted to be 'derived from the optimal control problem' while Appendix B.4 describes a state-dependent heuristic, so the benchmark is partly self-defined; nevertheless, the comparison is still against an explicitly specified rule rather than against FlowOE's own fitted output. The only identifiable self-citation is reference [16], used to motivate the finance transfer; it does not carry the derivation. No uniqueness theorem, imported ansatz, or renamed known result is load-bearing. Score 2 reflects one minor, non-load-bearing self-citation; the absent refinement mechanism should be treated as a correctness/completeness concern, not as circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central empirical results rest on a simulated Heston market whose experts include a heuristic strategy invented for this paper, Eq. 17, and PPO agents trained in the same simulator. The claimed refining loss, which the abstract calls a key innovation, appears nowhere in the training algorithm. The model is evaluated in the same environment family that generated its demonstrations, so the reported improvements are in-distribution gains. These choices load the ledger with domain assumptions and ad hoc constructs rather than externally validated benchmarks.

free parameters (3)

Heston-Optimal volatility adjustment exponent = 0.5
In Eq. 17, the factor with the square root of the expected future variance over current variance uses a hand-picked exponent of 0.5 to scale the trading rate; no derivation is provided.
Heston-Optimal base trading rate exponent = 1/(1+beta)
The term (x/(T-t))^(1/(1+beta)) in Eq. 17 and Appendix B.4 is an ad hoc adjustment for nonlinear impact; no derivation is provided.
Urgency factor cap in Appendix B.4 = Omega(tau)=min(2,1/tau)
The constant 2 and the reciprocal form are chosen by hand to bound trading urgency near maturity.

assumptions (5)

domain assumption The Heston stochastic volatility model with Milstein discretization accurately simulates the market dynamics for evaluating execution strategies.
The entire dataset and evaluation rest on this simulator, described in Section 5.1 and Appendix B.5. No real market data is used.
domain assumption Temporary market impact is well described by the power law f(nu)=epsilon times |nu|^beta with beta in (0,1].
Used in Eq. 7 and throughout; the paper notes empirical support for concavity but assumes this specific functional form.
ad hoc to paper The Heston-Optimal trading rate, Eq. 17, approximates the solution of the optimal control problem.
The formula is presented as 'derived from the optimal control problem' but no derivation or citation is given; it appears to be a heuristic constructed for this paper.
ad hoc to paper A refining loss or fine-tuning process exists in the FlowOE training pipeline and is responsible for the improvement over expert actions.
Announced in the abstract and Section 1, but never specified in Algorithm 1 or any other equation; the reported gains are attributed to it without evidence.
domain assumption The PPO agents and traditional models provide diverse, high-quality expert demonstrations for imitation learning.
The training corpus is generated from these experts, as described in Section 5.2 and Appendix B.2; the quality and diversity of these demonstrations is not independently validated.

invented entities (2)

Heston-Optimal trading strategy, Eq. 17
purpose: Used as a calibrated benchmark expert and as a source of demonstrations for the imitation model.
The formula is introduced in this paper without derivation, citation, or out-of-sample validation; it is a new heuristic whose optimality is assumed.
Refining loss or fine-tuning adjustment mechanism
purpose: Claimed to let the model search for an adjustment action that improves on the imitation output.
Mentioned as a key innovation in the abstract and Section 1, but never defined; no equation, algorithm, or ablation supports its existence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts." pith.science (2026). https://pith.science/paper/OFF4QUTK

@misc{pith2026250605755,
  author       = {Pith},
  title        = {Pith review of: FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFF4QUTK}},
  note         = {Machine review of arXiv:2506.05755}
}

read the original abstract

Optimal execution in financial markets refers to the process of strategically transacting a large volume of assets over a period to achieve the best possible outcome by balancing the trade-off between market impact costs and timing or volatility risks. Traditional optimal execution strategies, such as static Almgren-Chriss models, often prove suboptimal in dynamic financial markets. This paper propose flowOE, a novel imitation learning framework based on flow matching models, to address these limitations. FlowOE learns from a diverse set of expert traditional strategies and adaptively selects the most suitable expert behavior for prevailing market conditions. A key innovation is the incorporation of a refining loss function during the imitation process, enabling flowOE not only to mimic but also to improve upon the learned expert actions. To the best of our knowledge, this work is the first to apply flow matching models in a stochastic optimal execution problem. Empirical evaluations across various market conditions demonstrate that flowOE significantly outperforms both the specifically calibrated expert models and other traditional benchmarks, achieving higher profits with reduced risk. These results underscore the practical applicability and potential of flowOE to enhance adaptive optimal execution.

Figures

Figures reproduced from arXiv: 2506.05755 by the authors.

**Figure 2.** Shortcut Policy Learning the HO Experts 22 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 3.** Shortcut Policy Learning the PPO Experts [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

27 extracted references · 19 canonical work pages

[1]

Optimal execution of portfolio transac- tions

Almgren, R., Chriss, N., 2001. Optimal execution of portfolio transac- tions. Journal of Risk 3, 5–39

work page 2001
[2]

Optimalexecutionwithnonlinearimpactfunctions and trading-enhanced risk

Almgren, R.F., 2003. Optimalexecutionwithnonlinearimpactfunctions and trading-enhanced risk. Applied mathematical finance 10, 1–18

work page 2003
[3]

To hedge or not to hedge: Optimal strategies for stochastic trade flow management

Bergault, P., Guéant, O., Bodor, H., 2025. To hedge or not to hedge: Optimal strategies for stochastic trade flow management. arXiv preprint arXiv:2503.02496

arXiv 2025
[4]

Staticversusadaptedoptimalexecutionstrate- gies in two benchmark trading models, in: Innovations in Insurance, Risk-and Asset Management

Brigo, D., Piat, C., 2019. Staticversusadaptedoptimalexecutionstrate- gies in two benchmark trading models, in: Innovations in Insurance, Risk-and Asset Management. World Scientific, pp. 239–273

work page 2019
[5]

Openai gym

Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schul- man, J., Tang, J., Zaremba, W., 2016. Openai gym. URL: https://arxiv.org/abs/1606.01540,arXiv:1606.01540

arXiv 2016
[6]

Optimal trading with predictable return and stochastic volatility

Chan, P., Sircar, R., 2015. Optimal trading with predictable return and stochastic volatility. Available at SSRN 2623747

work page 2015
[7]

Optimal execution with uncertain order fills in almgren–chriss framework

Cheng, X., Di Giacinto, M., Wang, T.H., 2017. Optimal execution with uncertain order fills in almgren–chriss framework. Quantitative Finance 17, 55–69

work page 2017
[8]

Optimal execution with non- linear transient market impact

Curato, G., Gatheral, J., Lillo, F., 2017. Optimal execution with non- linear transient market impact. Quantitative Finance 17, 41–54

work page 2017

Show all 27 references

[9]

Optimalexecution: Areview

Donnelly, R., 2022. Optimalexecution: Areview. AppliedMathematical Finance 29, 181–212

2022
[10]

One step diffusion via shortcut models

Frans, K., Hafner, D., Levine, S., Abbeel, P., 2024. One step diffusion via shortcut models. arXiv preprint arXiv:2410.12557

2024 arXiv
[11]

Optimal trade execution under geometric brownian motion in the almgren and chriss framework

Gatheral, J., Schied, A., 2011. Optimal trade execution under geometric brownian motion in the almgren and chriss framework. International Journal of Theoretical and Applied Finance 14, 353–368

2011
[12]

Permanent market impact can be nonlinear

Guéant, O., 2013. Permanent market impact can be nonlinear. arXiv preprint arXiv:1305.0413 . 34

2013 arXiv
[13]

Optimal execution with reinforcement learn- ing

Hafsi, Y., Vittori, E., 2024. Optimal execution with reinforcement learn- ing. arXiv preprint arXiv:2411.06389

2024
[14]

Correlated order flow: Pervasiveness, sources, and pricing effects

Harford, J., Kaul, A., 2005. Correlated order flow: Pervasiveness, sources, and pricing effects. Journal of Financial and Quantitative Anal- ysis 40, 29–55

2005
[15]

Hendricks, D., Wilcox, D., 2014. A reinforcement learning extension to the almgren-chriss framework for optimal trade execution, in: 2014 IEEE Conference on computational intelligence for financial engineering & economics (CIFEr), IEEE. pp. 457–464

2014
[16]

Flowhft: Flow policy induced optimal high-frequency trading under diverse market conditions

Li, Y., Chen, Z., Yang, S., 2025. Flowhft: Flow policy induced optimal high-frequency trading under diverse market conditions. arXiv preprint arXiv:2505.05784

2025 arXiv
[17]

Flow matching for genera- tive modeling

Lipman, Y., Hagaj, R., Cohen-Or, D., 2022. Flow matching for genera- tive modeling. arXiv preprint arXiv:2210.02747

2022 arXiv
[18]

Rectified flow: A marginal preserving approach to optimal transport

Liu, Q., 2022. Rectified flow: A marginal preserving approach to optimal transport. arXiv preprint arXiv:2209.14577

2022 arXiv
[19]

Mean–variance optimal adaptive execu- tion

Lorenz, J., Almgren, R., 2011. Mean–variance optimal adaptive execu- tion. Applied Mathematical Finance 18, 395–422

2011
[20]

A class of optimal liquidation problem with a nonlinear temporary market impact

Ma, J., Gao, D., 2020. A class of optimal liquidation problem with a nonlinear temporary market impact. Mathematical Problems in Engi- neering 2020, 6614177

2020
[21]

Reinforcement learning for optimal execution when liquidity is time-varying

Macrì, A., Lillo, F., 2025. Reinforcement learning for optimal execution when liquidity is time-varying. Applied Mathematical Finance , 1–31

2025
[22]

Heston’s stochastic volatility model: Implementation, calibration and some extensions

Mikhailov, S., Nögel, U., 2004. Heston’s stochastic volatility model: Implementation, calibration and some extensions. John Wiley and Sons

2004
[23]

Optimal trading strategy and sup- ply/demand dynamics

Obizhaeva, A.A., Wang, J., 2013. Optimal trading strategy and sup- ply/demand dynamics. Journal of Financial markets 16, 1–32

2013
[24]

Proximal policy optimization algorithms.arXiv:1707.06347

Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O., 2017. Proximal policy optimization algorithms.arXiv:1707.06347. arXiv preprint arXiv:1707.06347. 35

2017 arXiv
[25]

Optimal execution: Linear market impact with ex- ponential decay

Skachkov, I., 2009. Optimal execution: Linear market impact with ex- ponential decay. Available at SSRN 2283027

2009
[26]

Consistency models

Song, Y., Dhariwal, P., Chen, M., Sutskever, I., 2023. Consistency models

2023
[27]

Conditional flow matching: Simulation- free dynamic optimal transport

Tong, A., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Fatras, K., Wolf, G., Bengio, Y., 2023. Conditional flow matching: Simulation- free dynamic optimal transport. arXiv preprint arXiv:2302.00482 2. 36

2023 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Optimal execution of portfolio transac- tions

Almgren, R., Chriss, N., 2001. Optimal execution of portfolio transac- tions. Journal of Risk 3, 5–39

work page 2001

[2] [2]

Optimalexecutionwithnonlinearimpactfunctions and trading-enhanced risk

Almgren, R.F., 2003. Optimalexecutionwithnonlinearimpactfunctions and trading-enhanced risk. Applied mathematical finance 10, 1–18

work page 2003

[3] [3]

To hedge or not to hedge: Optimal strategies for stochastic trade flow management

Bergault, P., Guéant, O., Bodor, H., 2025. To hedge or not to hedge: Optimal strategies for stochastic trade flow management. arXiv preprint arXiv:2503.02496

arXiv 2025

[4] [4]

Staticversusadaptedoptimalexecutionstrate- gies in two benchmark trading models, in: Innovations in Insurance, Risk-and Asset Management

Brigo, D., Piat, C., 2019. Staticversusadaptedoptimalexecutionstrate- gies in two benchmark trading models, in: Innovations in Insurance, Risk-and Asset Management. World Scientific, pp. 239–273

work page 2019

[5] [5]

Openai gym

Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schul- man, J., Tang, J., Zaremba, W., 2016. Openai gym. URL: https://arxiv.org/abs/1606.01540,arXiv:1606.01540

arXiv 2016

[6] [6]

Optimal trading with predictable return and stochastic volatility

Chan, P., Sircar, R., 2015. Optimal trading with predictable return and stochastic volatility. Available at SSRN 2623747

work page 2015

[7] [7]

Optimal execution with uncertain order fills in almgren–chriss framework

Cheng, X., Di Giacinto, M., Wang, T.H., 2017. Optimal execution with uncertain order fills in almgren–chriss framework. Quantitative Finance 17, 55–69

work page 2017

[8] [8]

Optimal execution with non- linear transient market impact

Curato, G., Gatheral, J., Lillo, F., 2017. Optimal execution with non- linear transient market impact. Quantitative Finance 17, 41–54

work page 2017