Paper: arXiv 2411.10726

Authors: Yan Dolinsky

Abstract

We study an optimal execution problem in the infinite horizon setup. Our financial market is given by the Black-Scholes model with a linear price impact. The main novelty of the current note is that we study the constrained case where the number of shares and the selling rate are non-negative processes. For this case we give a complete characterization of the value and the optimal control via a solution of a non-linear ordinary differential equation (ODE). Furthermore, we provide an example where the non-linear ODE can be solved explicitly. Our approach is purely probabilistic.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous theoretical extension to optimal execution under non-negativity constraints using probabilistic methods and non-linear ODEs, featuring an explicit solution via Bessel functions. However, it is entirely theoretical with zero empirical data or backtesting validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Execution with Monotone Strategies in BS Model] --> B[Model Setup: Black-Scholes + Linear Price Impact]
    B --> C[Constraint Application: Non-Negative Shares & Selling Rates]
    C --> D[Probabilistic Approach: Dynamic Programming & Verification]
    D --> E[Derivation of Non-Linear ODE]
    E --> F[Explicit Solution of ODE for Specific Example]
    F --> G[Outcomes: Complete Characterization of Value Function & Optimal Control]