Fredholm Approach to Nonlinear Propagator Models

ArXiv ID: 2503.04323 “View on arXiv”

Authors: Unknown

Abstract

We formulate and solve an optimal trading problem with alpha signals, where transactions induce a nonlinear transient price impact described by a general propagator model, including power-law decay. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear stochastic Fredholm equation with both forward and backward coefficients. We prove the existence and uniqueness of the solution under a monotonicity condition reflecting the nonlinearity of the price impact. Moreover, we derive an existence result for the optimal strategy beyond this condition when the underlying probability space is countable. In addition, we introduce a novel iterative scheme and establish its convergence to the optimal trading strategy. Finally, we provide a numerical implementation of the scheme that illustrates its convergence, stability, and the effects of concavity on optimal execution strategies under exponential and power-law decay.

Keywords: optimal execution, transient price impact, nonlinear stochastic control, propagator models, variational methods

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper centers on advanced stochastic analysis (Fredholm equations, variational methods, existence/uniqueness proofs), which is mathematically dense and theoretical. While it includes a numerical scheme to illustrate convergence and stability, it lacks real backtesting, dataset performance metrics, or implementation details for live trading, keeping empirical rigor low.
  flowchart TD
    A["Research Goal: Optimal Trading with<br/>Nonlinear Transient Price Impact"] --> B["Formulate Optimization Problem<br/>Variational Approach"]
    B --> C["Mathematical Analysis<br/>Prove Existence & Uniqueness<br/>via Fredholm Equations"]
    B --> D["Develop Numerical Scheme<br/>Iterative Solver"]
    C --> E["Key Finding: Solvability<br/>under Monotonicity Condition"]
    D --> F["Key Finding: Convergent<br/>Iterative Algorithm"]
    E & F --> G["Numerical Implementation<br/>Simulate Exponential &<br/>Power-Law Decay Strategies"]