On Bellman equation in the limit order optimization problem for high-frequency trading

ArXiv ID: 2510.15988 “View on arXiv”

Authors: M. I. Balakaeva, A. Yu. Veretennikov

Abstract

An approximation method for construction of optimal strategies in the bid & ask limit order book in the high-frequency trading (HFT) is studied. The basis is the article by M. Avellaneda & S. Stoikov 2008, in which certain seemingly serious gaps have been found; in the present paper they are carefully corrected. However, a bit surprisingly, our corrections do not change the main answer in the cited paper, so that, in fact, the gaps turn out to be unimportant. An explanation of this effect is offered.

Keywords: Limit order book, High-frequency trading, Optimal strategy construction, Avellaneda-Stoikov model, Stochastic control, General (High-Frequency Trading)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper centers on sophisticated mathematical derivations—solving a modified Bellman equation via asymptotic series in a controlled Bachelier/Poisson limit order model—making it dense in advanced stochastic calculus and PDE theory. However, it presents no real data, backtests, or implementation details; it is purely theoretical with a focus on correcting a prior paper’s gaps, thus lacking empirical validation.

  flowchart TD
    A["Research Goal<br>Derive optimal bid/ask<br>limit order strategies in HFT"] --> B["Methodology<br>Review & Correct Avellaneda-Stoikov<br>2008 model gaps"]
    B --> C["Data/Inputs<br>Limit Order Book<br>Stochastic Market Dynamics"]
    C --> D["Computational Process<br>Solve corrected<br>Bellman equation via<br>Stochastic Control"]
    D --> E{"Outcome"}
    E --> F["Key Finding 1<br>Corrections identified in<br>original model"]
    E --> G["Key Finding 2<br>Main optimal strategy result<br>remains unchanged"]
    E --> H["Key Finding 3<br>Explanation for why gaps<br>were mathematically unimportant"]