Hawkes-Driven OTC Market Making: Volterra-Riccati Approximation

Alexander Barzykin

REVIEW 4 major objections 3 minor 30 references

Hawkes-Driven OTC Market Making: Volterra-Riccati Approximation

T0 review · 4 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper shows that a state-feedback Volterra–Riccati quote rule, driven by the conditional forecast of future RFQ flow, closely approximates the exact optimal market-making policy for Hawkes-distributed requests and generates an endogenou

desk verdict A solid approximation paper for Hawkes-driven OTC market making, with a strong exponential benchmark and an honest but unvalidated long-memory section—worth refereeing. read the letter →

arxiv 2608.02002 v1 pith:34X7222N submitted 2026-08-03 q-fin.RM q-fin.MF

Alexander Barzykin This is my paper · ORCID

classification q-fin.RMq-fin.MF MSC 91G8060G5593E20

keywords OTCmarketmakingHawkesprocessesVolterra–RiccatiapproximationRFQflowimpactstochasticcontrollongmemoryoptimalquoting

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

OTC market makers receive request-for-quote (RFQ) arrivals whose clustering and persistence contain information about future client demand. This paper argues that a low-dimensional quote rule built from the conditional Volterra forecast of future RFQ intensity can capture most of the value of that information. The author develops a hierarchy of Volterra–Riccati approximations, culminating in a state-feedback rule that updates the forecast after each observed request, and validates it against an exact lifted HJB benchmark in an exponential Hawkes model. The state-feedback rule tracks the exact optimal quotes with small relative regret, especially when flow is directional, whereas a memory-free Poisson policy loses up to about 27% of the objective. In a long-memory power-law model, a directional burst becomes a persistent quote skew via the continuation-value shadow price, an endogenous OTC impact that reduces inventory and P&L risk.

What carries the argument

The machinery is the Volterra forecast curve m_t(u) = E[λ_{t+u} | F_t], a deterministic curve that summarizes the Hawkes memory. A hierarchy of approximations is built on it: the mean Volterra–Riccati surrogate solves a backward Riccati system with m_t as the driving intensity; the noise-aware version adds a covariance correction; the state-feedback version replaces m_t with the post-request forecast m_t + ρ_i, using the functional sensitivity of the value map to form the post-request shadow price. Quotes are always produced by the exact Hamiltonian optimizer applied to the approximated shadow price. In the exponential benchmark the forecast curve is represented by the finite-dimensional Haw

What would settle it

Solve the power-law-like Hawkes model with a high-fidelity numerical method (e.g., a very large exponential mixture lift, or a path-dependent PDE solver) and compare the resulting optimal quotes and value against the state-feedback Volterra–Riccati policy; if the relative regret grows with the kernel tail exponent, the approximation's long-memory claims fail. Alternatively, from real RFQ data, measure the decay of quote skew after a directional burst and check whether it matches the fitted kernel's tail exponent.

Watch

Extended reading notes

Core claim

The central discovery is that the path-dependent Hawkes market-making problem, which needs the full RFQ history as state, can be reduced to a tractable quote rule operating on the conditional forecast curve of future request intensities. The paper proves numerically, in an exponential Hawkes benchmark where the exact lifted HJB can be solved, that the state-feedback Volterra–Riccati policy — which updates the forecast curve after each request and recomputes the quadratic continuation value — closely tracks the exact optimal quotes, with relative regret around 0.08%–0.17% across the tested regimes, while a Poisson policy that ignores memory loses up to 27%. The same rule, when applied to a po

Load-bearing premise

The receding-horizon conditional surrogate is assumed to remain an accurate approximation of the true path-dependent value function for general long-memory kernels, but exact validation is only supplied for exponential kernels, and Section 7 explicitly states that a general error or convergence theory remains open.

Editorial extensions

If this is right

Ignoring request-flow history when flow is directional leads to systematically misplaced quotes and larger inventory risk, even if the dealer solves the misspecified Poisson problem optimally.
The state-feedback Volterra–Riccati rule provides a closed, low-dimensional quoting algorithm that remains applicable when exact Markovian lifting is impractical, such as power-law or very high-dimensional mixture kernels.
An observed directional RFQ burst should alter quotes persistently, not just momentarily, because the conditional forecast of future flow is itself persistent; the quote skew then decays with the forecast response tail.
Conditioning on the RFQ memory primarily improves risk control (reduces inventory and P&L variance), more than it changes mean P&L, in the long-memory experiment.

Reading between the lines

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

One could use the linearized impact formula (50) to invert observed quote-skew decay into an estimate of the Hawkes memory kernel, giving a new empirical test on RFQ data.
The approximation hierarchy suggests a modular production design: a Hawkes/Volterra forecast engine, a Riccati solver, and a tabulated Hamiltonian optimizer can replace a full dynamic-programming solver at low latency.
If the forecast response decays as a power law, the model predicts OTC quote impact should also decay as a power law with the same exponent; this is testable with dealer-level or platform-level RFQ data.
The method could be extended to treat win probabilities as state-dependent (e.g., competition or toxicity), but then the state-feedback update would need to propagate through the response function as well; the paper's validation does not cover that case.

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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

Short version: this paper earns a referee. The genuinely new piece is the state-feedback Volterra–Riccati quote rule for the OTC request/fill separation—the dealer updates the continuation-value shadow price with the post-request forecast curve, not just current intensity. I don't know that combination in the prior literature, and the paper situates it correctly against Jusselin and Bergault et al.

The exponential benchmark is the strongest part. Solving the exact lifted HJB on a grid and measuring paired regret against it is the right test. The Poisson comparator is itself an exact HJB solution within the wrong model, so the regret numbers cleanly separate misspecification error from approximation error. The near-critical and directional regimes show the mean forecast does a lot, and state feedback closes most of the gap. That is a real, reproducible numerical result.

The soft spot is Section 6. There is no exact benchmark for the power-law-like mixture, no convergence bound, and the paper admits in Section 7 that the conditional forecast surrogate does not satisfy a dynamic-programming principle and that a general error theory is open. On top of that, the persistent quote impact is largely constructed: the state-feedback shadow price builds the Volterra response rho_i into the quote, so the long-memory tail of the quote skew is inherited from the kernel by design. The inventory and P&L improvements over a Poisson dealer are consistent with conditioning on information, but they are not evidence that the approximate policy is near the true optimum in that regime. The exponential validation is one-dimensional and mean-reverting; it should not be oversold as certification for an 8-factor slowly decaying mixture. This is an addressable limitation—more tests, even a small exact lift or a different approximation, would help—but Section 6 should not be framed as validation.

The empirical motivation is honest: proprietary HSBC data, no error bars, two-way activity rather than signed flow, and the paper says so. That is acceptable for motivation, but keep it in that role.

Overall, the hierarchy is clearly formulated and the benchmark is well executed. The citation pattern is appropriate. I'd send it to a serious referee, with the instruction to focus on whether Section 6's claims are appropriately qualified. If the authors add a long-memory sanity check or soften the framing, this would be a publishable contribution.

Referee Report

4 major / 3 minor

Summary. The paper develops a Volterra–Riccati approximation hierarchy for OTC market making when RFQ arrivals are modelled by general Hawkes kernels. It sets up an exact path-dependent HJB, then replaces the history state with conditional future-intensity forecast curves and uses a quadratic/Riccati expansion, adding a covariance correction and a state-feedback update after observed RFQs. In an exponential Hawkes benchmark, the state-feedback policy is shown numerically to closely track the exact lifted HJB, while a memory-free Poisson policy suffers large regret. The same rule is then applied to a power-law-like N=8 exponential mixture; a directional RFQ burst produces a persistent quote skew and reduces inventory/P&L risk relative to a Poisson benchmark. The paper explicitly states that no general error or convergence theory is provided and that the frozen conditional forecast does not satisfy an intertemporal dynamic-programming principle.

Significance. The exponential validation is a genuine strength: the exact Markovian lift is solved and compared with common random numbers and paired regrets, and the state-feedback mechanism is shown to matter, especially in directional regimes. If the approximation can be trusted for long-memory kernels, the paper offers a tractable bridge between persistent RFQ flow and market-making controls. However, the headline long-memory application is not yet supported to the same standard. The quote-impact tail in Eq. (53) is inherited by construction from the forecast response in the approximate rule, and Section 6.2 lacks any non-Poisson reference solution; these gaps must be addressed before the long-memory claims can be accepted.

major comments (4)

[Section 6.2, Section 7] The power-law experiment cannot support the central long-memory claim because the only comparator is the Poisson policy. Section 6.2 states that no lifted HJB is solved for the N=8 exponential mixture, and Section 7 concedes that 'a general error or convergence theory remains open' and that the frozen conditional forecast does not satisfy an intertemporal dynamic-programming principle. Consequently Figs. 5–7 establish properties of the approximate state-feedback rule, not of the exact optimal policy; an approximate policy can beat a misspecified Poisson policy while being far from the true path-dependent optimum. Please add a reference benchmark or error calibration in this regime, e.g., an exact lifted HJB for a smaller exponential mixture (N=2 or N=3) under the same economic parameters, a convergence study in N_f and N, or an a posteriori bound using the covariance/curvature terms in S
[Section 6.1, Eqs. (50)–(53)] The long-memory impact tail is hardwired into the approximate rule. Eq. (50) defines the incremental shadow-price impact as a finite-lag convolution of the resolvent response rho_{i0} with the sensitivity difference D(t,q;u)-D(t,q+epsilon_i z_i;u). If that sensitivity kernel is integrable, the convolution automatically preserves the tail order of rho, so Eq. (53) is a mathematical consequence of the definition of the state-feedback rule, not a testable prediction of the exact optimal policy. The abstract's wording, 'endogenous OTC quote impact inherits the long-memory decay of the RFQ forecast response', is therefore accurate for the approximate policy, but it should be labelled as such; an independent optimality argument would be needed to claim that the true optimal policy has this property.
[Section 4.3, Eqs. (36)–(38), (42), (50)] The noise-aware correction and the linearized impact formulas require twice functional differentiability of the map m -> V0(t,q;m), which is assumed but not proved or even stated with conditions. Since V0 is obtained from a Riccati system with coefficients linear in m, the derivatives are likely available under mild integrability conditions, but the paper does not derive the resulting sensitivity ODEs or state the required function space for m. This matters for Eq. (50), where integrability in u of the sensitivity difference is precisely what allows the tail to be pulled through the convolution. Please state the assumptions and verify the derivative equations used in the implementation, at least in the exponential benchmark where the exact lifted HJB is available.
[Section 5.1, numerical truncation] The validation relies on truncated inventory and memory grids (qmax=50, Xmax=800, 1000 memory points) but no grid-convergence or truncation-sensitivity study is reported. Since the exact benchmark is itself discretized, the claims about 0.08–0.17% relative regret for the state-feedback policy should be accompanied by a check that the discretization is fine enough and that the truncation does not bias the comparison. This is not a fatal issue, but it is needed to make the numerical validation fully convincing.

minor comments (3)

[Section 2, Table 1] The fitted branching ratios and mixture weights are reported without standard errors, confidence intervals, or goodness-of-fit diagnostics, and the data are proprietary. Since Table 1 is the empirical motivation for the entire model, even a brief indication of estimation uncertainty would help the reader assess the strength of the persistence evidence.
[Section 5.1 and Section 6.2] There are small notational inconsistencies in the sigmoid win-probability formulas: the parameterization in Section 5.1 (f(δ) = (1+exp(δ-1.2))^{-1}) and in Section 6.2 (f(δ) = (1+exp(3(δ-1)))^{-1}) differ, and the displayed formula in Section 5.1 has a parenthesis imbalance. These should be cleaned up.
[Section 7] The paper is commendably explicit about its limitations, but the concluding section could more directly state that the power-law results in Section 6 are properties of the approximate policy, not of the exact solution of the path-dependent HJB, unless the proposed convergence/benchmark work is added.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the power-law impact tail is an explicitly derived model implication, and the exponential benchmark provides an independent validation reference; the only self-citation is non-load-bearing.

full rationale

The derivation chain is self-contained. The exact path-dependent HJB, Eq. (18), is a genuine optimal-control formulation, and the Volterra–Riccati objects in Eqs. (23)–(43) are explicitly labeled as conditional surrogates: the paper states, e.g., in Section 4.2, that Eq. (23) 'is not claimed to be the HJB equation of the original Hawkes control problem, nor a time-consistent auxiliary value function.' The exponential benchmark in Section 5 is a meaningful external check: the state-feedback rule is compared against a numerically solved lifted HJB, which is not constructed from the approximation, and the reported closeness is a nontrivial numerical result. The power-law impact section is an analytic consequence rather than a fitted prediction. Section 6.1 defines the incremental shadow-price impact, Eq. (50), as a Volterra filter of the resolvent response rho_i, and then derives the tail behavior in Eqs. (52)–(53). The text explicitly says 'the impact is therefore a Volterra filter of the RFQ forecast response' and that 'the time decay is inherited from the RFQ-memory kernel.' This is an openly derived model implication, not an empirical result that has been fitted to data and then re-reported as a discovery. The Poisson-benchmark comparison in Section 6.2 is a controlled numerical illustration of the value of conditioning, not a hidden circular step. The only self-citation (Barzykin 2026) is a passing literature contrast — 'complementary to the inventory-based impact of a trade' — and is not load-bearing. Section 7's concession that 'a general error or convergence theory remains open' is an honest limitation on the approximation, not a circularity. No circular step meeting the evidence threshold was found.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical or model entities are postulated; the conditional forecast curve m_t(u) and forecast response rho_i are mathematical constructions within the model, not independent entities with falsifiable handles outside the paper.

free parameters (5)

Fitted Hawkes branching ratios and mixture weights = eta 0.862-0.895; weights w1, w10, w60 (Table 1)
Fitted to proprietary HSBC spot-FX RFQ data; used only as empirical motivation, not in the control experiments.
Power-law kernel mixture (branching matrix and exponent) = B = [[0.55,0.05],[0.05,0.55]], nu_mix=0.65, N=8, log-spaced decay rates
Chosen by hand to represent a slowly decaying power-law-like memory with spectral radius 0.6; central to Section 6 impact experiment.
Win probability sigmoid parameters = Benchmark: slope 1, offset 1.2; power-law: slope 3, offset 1
Model choices for fill probability; not estimated from data.
Risk aversion and terminal penalty = Benchmark kappa=0.2, kappa_T=0.2; power-law kappa=2.0, kappa_T=0.01
Chosen to control inventory risk; standard in market-making models.
Horizon and discretization parameters = T=1 day (benchmark), Teval=0.1, Tcont=0.1, grid sizes as in Section 5.1
Numerical implementation parameters; not fitted to data.

assumptions (6)

domain assumption The decomposition of the dealer objective into spread revenue minus quadratic inventory risk (14) is the correct reduced control objective under martingale mid-price.
Standard in Avellaneda-Stoikov/Ho-Stoll literature; assumes risk aversion is quadratic and terminal penalty is given.
domain assumption Request arrivals are exogenous and independent of the dealer's quotes; fills do not feed back into future RFQ intensity.
Central modeling convention (Section 3.1, Eqs. 7-10); if quotes affect request creation, the conditional forecast state is endogenous and the whole state-feedback rule changes. Relaxation mentioned in Section 7.
ad hoc to paper The quadratic Hamiltonian expansion (25) and quadratic inventory ansatz (26) accurately approximate the continuation value.
Inherited from Bergault et al. (2021); used in the Riccati system (29)-(31). No error bound is provided.
ad hoc to paper The map from forecast curve m to mean Volterra-Riccati value V0 is twice functionally differentiable (36).
Needed for the noise-aware correction (38); not proven.
ad hoc to paper For power-law kernels, the Riccati sensitivity kernel in (50) is integrable and finite-horizon effects are negligible so the impact tail matches the Hawkes response tail (52)-(53).
Explicitly stated as a long-horizon approximation in Section 6.1.
ad hoc to paper Existence and uniqueness of the path-dependent HJB (18) is assumed.
Equation (18) is presented formally; the paper explicitly does not prove well-posedness for general kernels (Section 5.1).

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Cite this review

Pith. "Pith review of Hawkes-Driven OTC Market Making: Volterra-Riccati Approximation." pith.science (2026). https://pith.science/paper/34X7222N

@misc{pith2026260802002,
  author       = {Pith},
  title        = {Pith review of: Hawkes-Driven OTC Market Making: Volterra-Riccati Approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34X7222N}},
  note         = {Machine review of arXiv:2608.02002}
}

read the original abstract

We formulate an over-the-counter (OTC) market-making problem in which request-for-quote (RFQ) arrivals are modelled by general Hawkes kernels and fills are controlled thinnings of the exogenous request flow. The modelling choice is motivated by spot-FX RFQ data: after filtering and transforming to seasonality-adjusted RFQ activity time, two-way activity in major currency pairs exhibits large fitted branching ratios and multi-scale persistence. For general Hawkes kernels the control problem is path-dependent: the relevant state contains the order-flow history, or equivalently the forward curve of conditional future RFQ intensities. Exact Markovian lifting is available for exponential kernels, but it becomes high-dimensional for mixtures of exponentials and impractical for long-memory kernels. We therefore develop a hierarchy of Volterra-Riccati approximations. The first level replaces random future request flow by its conditional Volterra forecast; the second adds a covariance correction for intensity uncertainty; the third updates the quote rule with the realized Hawkes memory, or equivalently with the post-request conditional forecast curve. The approximation hierarchy is validated in an exponential Hawkes benchmark, where the exact lifted HJB can be solved numerically. The state-feedback Volterra-Riccati policy closely tracks the exact benchmark, especially in directional regimes, while a memory-free Poisson policy suffers substantial regret. We then apply the same state-feedback rule to a power-law-like RFQ memory model. A directional RFQ burst changes the conditional forecast of future flow and is converted by the continuation-value shadow price into a persistent quote skew. The resulting endogenous OTC quote impact inherits the long-memory decay of the RFQ forecast response and improves inventory and P&L risk control relative to a no-conditioning Poisson benchmark.

Figures

Figures reproduced from arXiv: 2608.02002 by the authors.

**Figure 2.** Approximation hierarchy. The modelling problem is Hawkes-driven and generally non-Markovian. The [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Relative paired regret against the exact lifted HJB across the three exponential Hawkes validation regimes. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Directional Hawkes benchmarks: ask offsets [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Power-law RFQ memory (conditional imbalance, left, log-log) and direct quote impact at zero inventory [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Inventory response after a directional RFQ burst. The Poisson dealer ignores the conditional future-flow [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Value of conditioning on the RFQ memory. As the initial directional burst becomes larger, the conditional [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 4 linked inside Pith

[1]

Annals of Applied Probability, 2021, 31 (5), 2244--2274

Abi Jaber, E., Miller, E., and Pham, H., Linear-quadratic control for a class of stochastic Volterra equations: solvability and approximation. Annals of Applied Probability, 2021, 31 (5), 2244--2274

2021
[2]

and Neuman, E., Optimal liquidation with signals: the general propagator case

Abi Jaber, E. and Neuman, E., Optimal liquidation with signals: the general propagator case. Mathematical Finance, 2025, 35 (4), 841--866

2025
[3]

and Samuel N

Allan, A.L. and Samuel N. Cohen, S.N., Pathwise stochastic control with applications to robust filtering. Annals of Applied Probability, 2020, 30 (5), 2274 -- 2310

2020
[4]

and Blanc, P., Dynamic optimal execution in a mixed-market-impact Hawkes price model

Alfonsi, A. and Blanc, P., Dynamic optimal execution in a mixed-market-impact Hawkes price model. Finance and Stochastics, 2016, 20 , 183--218

2016
[5]

and Stoikov, S., High-frequency trading in a limit order book

Avellaneda, M. and Stoikov, S., High-frequency trading in a limit order book. Quantitative Finance, 2008, 8 (3), 217--224

2008
[6]

Market Microstructure and Liquidity, 2015, 1 (1), 1550005

Bacry, E., Mastromatteo, I., and Muzy, J.-F., Hawkes processes in finance. Market Microstructure and Liquidity, 2015, 1 (1), 1550005

2015
[7]

and Keller, C., Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes

Bandini, E. and Keller, C., Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes. arXiv:2408.02147, 2024

arXiv 2024
[8]

SIAM Journal on Control and Optimization, 2025, 63 (5), 3189--3218

Bank, P., Bayer, C., Hager, P.P., Riedel, S., and Nauen, T., Stochastic Control with Signatures. SIAM Journal on Control and Optimization, 2025, 63 (5), 3189--3218

2025

Show all 30 references

[9]

arXiv:2601.13421, 2026

Barzykin, A., Market Making and Transient Impact in Spot FX. arXiv:2601.13421, 2026

2026 arXiv
[10]

and Vieira, D., Closed-form approximations in multi-asset market making

Bergault, P., Evangelista, D., Guéant, O. and Vieira, D., Closed-form approximations in multi-asset market making. Applied Mathematical Finance, 2021, 28 (2), 101--142

2021
[11]

and Gu\'eant, O., Liquidity Dynamics in RFQ Markets and Impact on Pricing

Bergault, P. and Gu\'eant, O., Liquidity Dynamics in RFQ Markets and Impact on Pricing. arXiv:2309.04216, 2023

2023 arXiv
[12]

arXiv:2405.03496, 2024

Bergault, P., Bertucci, L., Bouba, D., Gu\'eant, O., and Guilbert, J., Price-aware automated market makers: models beyond Brownian prices and static liquidity. arXiv:2405.03496, 2024

2024 arXiv
[13]

and Penalva, J., Algorithmic and High Frequency Trading , 2015 (Cambridge University Press: Cambridge, United Kingdom)

Cartea, \'A , Jaimungal, S. and Penalva, J., Algorithmic and High Frequency Trading , 2015 (Cambridge University Press: Cambridge, United Kingdom)

2015
[14]

SIAM Review, 2018, 60 (3), 673--703

Cartea, A., Jaimungal, S., and Ricci, J., Algorithmic trading, stochastic control, and mutually exciting processes. SIAM Review, 2018, 60 (3), 673--703

2018
[15]

and Bouchaud, J.-P., Price impact without order book: a study of the OTC credit index market

Eisler, Z. and Bouchaud, J.-P., Price impact without order book: a study of the OTC credit index market. arXiv:1609.04620, 2016

2016 arXiv
[16]

Guéant, O., The Financial Mathematics of Market Liquidity: From optimal execution to market making , 2016 (Chapman and Hall/CRC: Boca Raton, FL)

2016
[17]

and Fernandez-Tapia, J., Dealing with the inventory risk: a solution to the market making problem

Guéant, O., Lehalle, C.-A. and Fernandez-Tapia, J., Dealing with the inventory risk: a solution to the market making problem. Mathematics and Financial Economics, 2013, 7 , 477--507

2013
[18]

J., Bercot, N., and Bouchaud, J.-P., Critical reflexivity in financial markets: a Hawkes process analysis

Hardiman, S. J., Bercot, N., and Bouchaud, J.-P., Critical reflexivity in financial markets: a Hawkes process analysis. European Physical Journal B, 2013, 86 , 442

2013
[19]

and R\'eveillac, A., On the chaotic expansion for counting processes

Hillairet, C. and R\'eveillac, A., On the chaotic expansion for counting processes. Electronic Journal of Probability, 2024, 29 , 1--33

2024
[20]

and Stoll, H

Ho, T. and Stoll, H. R., Optimal dealer pricing under transactions and return uncertainty. Journal of Financial Economics, 1981, 9 (1), 47--73

1981
[21]

arXiv:2510.26438, 2025

Jain, K., Firoozye, N., Kochems, J., and Treleaven, P., An impulse control approach to market making in a Hawkes LOB market. arXiv:2510.26438, 2025

2025
[22]

Quantitative Finance, 2015, 15 (7), 1123--1135

Jaisson, T., Market impact as anticipation of the order flow imbalance. Quantitative Finance, 2015, 15 (7), 1123--1135

2015
[23]

and Rosenbaum, M., Limit theorems for nearly unstable Hawkes processes

Jaisson, T. and Rosenbaum, M., Limit theorems for nearly unstable Hawkes processes. Annals of Applied Probability, 2015, 25 (2), 600--631

2015
[24]

SIAM Journal on Financial Mathematics, 2021, 12 (3), 1150--1200

Jusselin, P., Optimal market making with persistent order flow. SIAM Journal on Financial Mathematics, 2021, 12 (3), 1150--1200

2021
[25]

and Rosenbaum, M., No-arbitrage implies power-law market impact and rough volatility

Jusselin, P. and Rosenbaum, M., No-arbitrage implies power-law market impact and rough volatility. Mathematical Finance, 2020, 30 (4), 1309--1336

2020
[26]

Journal of Banking and Financial Technology, 2024, 8 (1), 11--28

Kumar, P., Deep Hawkes process for high-frequency market making. Journal of Banking and Financial Technology, 2024, 8 (1), 11--28

2024
[27]

and Challet, D., The limits of statistical significance of Hawkes processes fitted to financial data

Lallouache, M. and Challet, D., The limits of statistical significance of Hawkes processes fitted to financial data. Quantitative Finance, 2016, 16 (1), 1--11

2016
[28]

Journal of Financial Markets, 2007, 10 (1), 1--25

Large, J., Measuring the resiliency of an electronic limit order book. Journal of Financial Markets, 2007, 10 (1), 1--25

2007
[29]

and Tse, A

Martin, N. and Tse, A. S. L., Optimal market-making with Hawkes process: a Markovian approximation approach via Mercer's expansion. SSRN.6451058, 2026

2026
[30]

arXiv:2601.23172, 2026

Muhle-Karbe, J., Ouazzani Chahdi, Y., Rosenbaum, M., and Szymanski, G., A unified theory of order flow, market impact, and volatility. arXiv:2601.23172, 2026

2026

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Annals of Applied Probability, 2021, 31 (5), 2244--2274

Abi Jaber, E., Miller, E., and Pham, H., Linear-quadratic control for a class of stochastic Volterra equations: solvability and approximation. Annals of Applied Probability, 2021, 31 (5), 2244--2274

2021

[2] [2]

and Neuman, E., Optimal liquidation with signals: the general propagator case

Abi Jaber, E. and Neuman, E., Optimal liquidation with signals: the general propagator case. Mathematical Finance, 2025, 35 (4), 841--866

2025

[3] [3]

and Samuel N

Allan, A.L. and Samuel N. Cohen, S.N., Pathwise stochastic control with applications to robust filtering. Annals of Applied Probability, 2020, 30 (5), 2274 -- 2310

2020

[4] [4]

and Blanc, P., Dynamic optimal execution in a mixed-market-impact Hawkes price model

Alfonsi, A. and Blanc, P., Dynamic optimal execution in a mixed-market-impact Hawkes price model. Finance and Stochastics, 2016, 20 , 183--218

2016

[5] [5]

and Stoikov, S., High-frequency trading in a limit order book

Avellaneda, M. and Stoikov, S., High-frequency trading in a limit order book. Quantitative Finance, 2008, 8 (3), 217--224

2008

[6] [6]

Market Microstructure and Liquidity, 2015, 1 (1), 1550005

Bacry, E., Mastromatteo, I., and Muzy, J.-F., Hawkes processes in finance. Market Microstructure and Liquidity, 2015, 1 (1), 1550005

2015

[7] [7]

and Keller, C., Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes

Bandini, E. and Keller, C., Non-local Hamilton-Jacobi-Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes. arXiv:2408.02147, 2024

arXiv 2024

[8] [8]

SIAM Journal on Control and Optimization, 2025, 63 (5), 3189--3218

Bank, P., Bayer, C., Hager, P.P., Riedel, S., and Nauen, T., Stochastic Control with Signatures. SIAM Journal on Control and Optimization, 2025, 63 (5), 3189--3218

2025