On the minimizers of energy forms with completely monotone kernel

TL;DR

This paper investigates the properties of minimizers for energy functionals with completely monotone kernels, revealing their analytic structure and equivalence to constrained minimization problems, with applications to optimal portfolio liquidation.

Contribution

It provides a detailed characterization of minimizers, showing they are analytic with power series expansions and establishing their equivalence to constrained energy minimization.

Findings

Minimizers are analytic functions with even power series expansions.

Minimizers have nonnegative coefficients in their power series.

The minimization problem is equivalent to a nonnegativity constrained problem.

Abstract

Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.