Fredholm Approach to Nonlinear Propagator Models

Eduardo Abi Jaber, Alessandro Bondi, Nathan de Carvalho, Eyal Neuman and Sturmius Tuschmann
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Eduardo Abi Jaber: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Alessandro Bondi: Dipartimento di Economia e Finanza [Roma] - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma]
Nathan de Carvalho: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Eyal Neuman: Imperial College London
Sturmius Tuschmann: Imperial College London

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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 trading; Nonlinear market impact; Propagator models; Power-law decay; Square root-law; Fredholm equations; Nonlinear stochastic control (search for similar items in EconPapers)
Date: 2025-03-06
Note: View the original document on HAL open archive server: https://hal.science/hal-04981003v1
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